Oliver T. Unke, Martin Stöhr, Stefan Ganscha et al. · deepmind
Molecular dynamics (MD) simulations allow atomistic insights into chemical and biological processes. Accurate MD simulations require computationally demanding quantum-mechanical calculations, being practically limited to short timescales and few atoms. For larger systems, efficient, but much less reliable empirical force fields are used. Recently, machine learned force fields (MLFFs) emerged as an alternative means to execute MD simulations, offering similar accuracy as ab initio methods at orders-of-magnitude speedup. Until now, MLFFs mainly capture short-range interactions in small molecules or periodic materials, due to the increased complexity of constructing models and obtaining reliable reference data for large molecules, where long-ranged many-body effects become important. This work proposes a general approach to constructing accurate MLFFs for large-scale molecular simulations (GEMS) by training on "bottom-up" and "top-down" molecular fragments of varying size, from which the relevant physicochemical interactions can be learned. GEMS is applied to study the dynamics of alanine-based peptides and the 46-residue protein crambin in aqueous solution, allowing nanosecond-scale MD simulations of >25k atoms at essentially ab initio quality. Our findings suggest that structural motifs in peptides and proteins are more flexible than previously thought, indicating that simulations at ab initio accuracy might be necessary to understand dynamic biomolecular processes such as protein (mis)folding, drug-protein binding, or allosteric regulation.
Ilia Sucholutsky, Lukas Muttenthaler, Adrian Weller et al. · berkeley, cambridge
Biological and artificial information processing systems form representations of the world that they can use to categorize, reason, plan, navigate, and make decisions. How can we measure the similarity between the representations formed by these diverse systems? Do similarities in representations then translate into similar behavior? If so, then how can a system's representations be modified to better match those of another system? These questions pertaining to the study of representational alignment are at the heart of some of the most promising research areas in contemporary cognitive science, neuroscience, and machine learning. In this Perspective, we survey the exciting recent developments in representational alignment research in the fields of cognitive science, neuroscience, and machine learning. Despite their overlapping interests, there is limited knowledge transfer between these fields, so work in one field ends up duplicated in another, and useful innovations are not shared effectively. To improve communication, we propose a unifying framework that can serve as a common language for research on representational alignment, and map several streams of existing work across fields within our framework. We also lay out open problems in representational alignment where progress can benefit all three of these fields. We hope that this paper will catalyze cross-disciplinary collaboration and accelerate progress for all communities studying and developing information processing systems.
Alexander Binder, Leander Weber, Sebastian Lapuschkin et al.
While the evaluation of explanations is an important step towards trustworthy models, it needs to be done carefully, and the employed metrics need to be well-understood. Specifically model randomization testing is often overestimated and regarded as a sole criterion for selecting or discarding certain explanation methods. To address shortcomings of this test, we start by observing an experimental gap in the ranking of explanation methods between randomization-based sanity checks [1] and model output faithfulness measures (e.g. [25]). We identify limitations of model-randomization-based sanity checks for the purpose of evaluating explanations. Firstly, we show that uninformative attribution maps created with zero pixel-wise covariance easily achieve high scores in this type of checks. Secondly, we show that top-down model randomization preserves scales of forward pass activations with high probability. That is, channels with large activations have a high probility to contribute strongly to the output, even after randomization of the network on top of them. Hence, explanations after randomization can only be expected to differ to a certain extent. This explains the observed experimental gap. In summary, these results demonstrate the inadequacy of model-randomization-based sanity checks as a criterion to rank attribution methods.
Lukas Muttenthaler, Klaus Greff, Frieda Born et al. · deepmind, stanford
Deep neural networks have achieved success across a wide range of applications, including as models of human behavior and neural representations in vision tasks. However, neural network training and human learning differ in fundamental ways, and neural networks often fail to generalize as robustly as humans do raising questions regarding the similarity of their underlying representations. What is missing for modern learning systems to exhibit more human-aligned behavior? We highlight a key misalignment between vision models and humans: whereas human conceptual knowledge is hierarchically organized from fine- to coarse-scale distinctions, model representations do not accurately capture all these levels of abstraction. To address this misalignment, we first train a teacher model to imitate human judgments, then transfer human-aligned structure from its representations to refine the representations of pretrained state-of-the-art vision foundation models via finetuning. These human-aligned models more accurately approximate human behavior and uncertainty across a wide range of similarity tasks, including a new dataset of human judgments spanning multiple levels of semantic abstractions. They also perform better on a diverse set of machine learning tasks, increasing generalization and out-of-distribution robustness. Thus, infusing neural networks with additional human knowledge yields a best-of-both-worlds representation that is both more consistent with human cognitive judgments and more practically useful, thus paving the way toward more robust, interpretable, and human-aligned artificial intelligence systems.
Philipp Liznerski, Lukas Ruff, Robert A. Vandermeulen et al.
Due to the intractability of characterizing everything that looks unlike the normal data, anomaly detection (AD) is traditionally treated as an unsupervised problem utilizing only normal samples. However, it has recently been found that unsupervised image AD can be drastically improved through the utilization of huge corpora of random images to represent anomalousness; a technique which is known as Outlier Exposure. In this paper we show that specialized AD learning methods seem unnecessary for state-of-the-art performance, and furthermore one can achieve strong performance with just a small collection of Outlier Exposure data, contradicting common assumptions in the field of AD. We find that standard classifiers and semi-supervised one-class methods trained to discern between normal samples and relatively few random natural images are able to outperform the current state of the art on an established AD benchmark with ImageNet. Further experiments reveal that even one well-chosen outlier sample is sufficient to achieve decent performance on this benchmark (79.3% AUC). We investigate this phenomenon and find that one-class methods are more robust to the choice of training outliers, indicating that there are scenarios where these are still more useful than standard classifiers. Additionally, we include experiments that delineate the scenarios where our results hold. Lastly, no training samples are necessary when one uses the representations learned by CLIP, a recent foundation model, which achieves state-of-the-art AD results on CIFAR-10 and ImageNet in a zero-shot setting.
Pattarawat Chormai, Jan Herrmann, Klaus-Robert Müller et al.
Explainable AI aims to overcome the black-box nature of complex ML models like neural networks by generating explanations for their predictions. Explanations often take the form of a heatmap identifying input features (e.g. pixels) that are relevant to the model's decision. These explanations, however, entangle the potentially multiple factors that enter into the overall complex decision strategy. We propose to disentangle explanations by extracting at some intermediate layer of a neural network, subspaces that capture the multiple and distinct activation patterns (e.g. visual concepts) that are relevant to the prediction. To automatically extract these subspaces, we propose two new analyses, extending principles found in PCA or ICA to explanations. These novel analyses, which we call principal relevant component analysis (PRCA) and disentangled relevant subspace analysis (DRSA), maximize relevance instead of e.g. variance or kurtosis. This allows for a much stronger focus of the analysis on what the ML model actually uses for predicting, ignoring activations or concepts to which the model is invariant. Our approach is general enough to work alongside common attribution techniques such as Shapley Value, Integrated Gradients, or LRP. Our proposed methods show to be practically useful and compare favorably to the state of the art as demonstrated on benchmarks and three use cases.
J. Thorben Frank, Oliver T. Unke, Klaus-Robert Müller
The application of machine learning methods in quantum chemistry has enabled the study of numerous chemical phenomena, which are computationally intractable with traditional ab-initio methods. However, some quantum mechanical properties of molecules and materials depend on non-local electronic effects, which are often neglected due to the difficulty of modeling them efficiently. This work proposes a modified attention mechanism adapted to the underlying physics, which allows to recover the relevant non-local effects. Namely, we introduce spherical harmonic coordinates (SPHCs) to reflect higher-order geometric information for each atom in a molecule, enabling a non-local formulation of attention in the SPHC space. Our proposed model So3krates - a self-attention based message passing neural network - uncouples geometric information from atomic features, making them independently amenable to attention mechanisms. Thereby we construct spherical filters, which extend the concept of continuous filters in Euclidean space to SPHC space and serve as foundation for a spherical self-attention mechanism. We show that in contrast to other published methods, So3krates is able to describe non-local quantum mechanical effects over arbitrary length scales. Further, we find evidence that the inclusion of higher-order geometric correlations increases data efficiency and improves generalization. So3krates matches or exceeds state-of-the-art performance on popular benchmarks, notably, requiring a significantly lower number of parameters (0.25 - 0.4x) while at the same time giving a substantial speedup (6 - 14x for training and 2 - 11x for inference) compared to other models.
Robustness has become an important consideration in deep learning. With the help of explainable AI, mismatches between an explained model's decision strategy and the user's domain knowledge (e.g. Clever Hans effects) have been identified as a starting point for improving faulty models. However, it is less clear what to do when the user and the explanation agree. In this paper, we demonstrate that acceptance of explanations by the user is not a guarantee for a machine learning model to be robust against Clever Hans effects, which may remain undetected. Such hidden flaws of the model can nevertheless be mitigated, and we demonstrate this by contributing a new method, Explanation-Guided Exposure Minimization (EGEM), that preemptively prunes variations in the ML model that have not been the subject of positive explanation feedback. Experiments demonstrate that our approach leads to models that strongly reduce their reliance on hidden Clever Hans strategies, and consequently achieve higher accuracy on new data.
Thomas Schnake, Farnoush Rezaei Jafari, Jonas Lederer et al.
Explainable Artificial Intelligence (XAI) plays a crucial role in fostering transparency and trust in AI systems, where traditional XAI approaches typically offer one level of abstraction for explanations, often in the form of heatmaps highlighting single or multiple input features. However, we ask whether abstract reasoning or problem-solving strategies of a model may also be relevant, as these align more closely with how humans approach solutions to problems. We propose a framework, called Symbolic XAI, that attributes relevance to symbolic queries expressing logical relationships between input features, thereby capturing the abstract reasoning behind a model's predictions. The methodology is built upon a simple yet general multi-order decomposition of model predictions. This decomposition can be specified using higher-order propagation-based relevance methods, such as GNN-LRP, or perturbation-based explanation methods commonly used in XAI. The effectiveness of our framework is demonstrated in the domains of natural language processing (NLP), vision, and quantum chemistry (QC), where abstract symbolic domain knowledge is abundant and of significant interest to users. The Symbolic XAI framework provides an understanding of the model's decision-making process that is both flexible for customization by the user and human-readable through logical formulas.
Oliver Eberle, Jochen Büttner, Hassan El-Hajj et al.
Historical materials are abundant. Yet, piecing together how human knowledge has evolved and spread both diachronically and synchronically remains a challenge that can so far only be very selectively addressed. The vast volume of materials precludes comprehensive studies, given the restricted number of human specialists. However, as large amounts of historical materials are now available in digital form there is a promising opportunity for AI-assisted historical analysis. In this work, we take a pivotal step towards analyzing vast historical corpora by employing innovative machine learning (ML) techniques, enabling in-depth historical insights on a grand scale. Our study centers on the evolution of knowledge within the `Sacrobosco Collection' -- a digitized collection of 359 early modern printed editions of textbooks on astronomy used at European universities between 1472 and 1650 -- roughly 76,000 pages, many of which contain astronomic, computational tables. An ML based analysis of these tables helps to unveil important facets of the spatio-temporal evolution of knowledge and innovation in the field of mathematical astronomy in the period, as taught at European universities.
Ann-Kathrin Dombrowski, Jan E. Gerken, Klaus-Robert Müller et al.
Counterfactuals can explain classification decisions of neural networks in a human interpretable way. We propose a simple but effective method to generate such counterfactuals. More specifically, we perform a suitable diffeomorphic coordinate transformation and then perform gradient ascent in these coordinates to find counterfactuals which are classified with great confidence as a specified target class. We propose two methods to leverage generative models to construct such suitable coordinate systems that are either exactly or approximately diffeomorphic. We analyze the generation process theoretically using Riemannian differential geometry and validate the quality of the generated counterfactuals using various qualitative and quantitative measures.
Deep Neural Networks (DNNs) excel at learning complex abstractions within their internal representations. However, the concepts they learn remain opaque, a problem that becomes particularly acute when models unintentionally learn spurious correlations. In this work, we present DORA (Data-agnOstic Representation Analysis), the first data-agnostic framework for analyzing the representational space of DNNs. Central to our framework is the proposed Extreme-Activation (EA) distance measure, which assesses similarities between representations by analyzing their activation patterns on data points that cause the highest level of activation. As spurious correlations often manifest in features of data that are anomalous to the desired task, such as watermarks or artifacts, we demonstrate that internal representations capable of detecting such artifactual concepts can be found by analyzing relationships within neural representations. We validate the EA metric quantitatively, demonstrating its effectiveness both in controlled scenarios and real-world applications. Finally, we provide practical examples from popular Computer Vision models to illustrate that representations identified as outliers using the EA metric often correspond to undesired and spurious concepts.
Niklas Frederik Schmitz, Klaus-Robert Müller, Stefan Chmiela
Reconstructing force fields (FFs) from atomistic simulation data is a challenge since accurate data can be highly expensive. Here, machine learning (ML) models can help to be data economic as they can be successfully constrained using the underlying symmetry and conservation laws of physics. However, so far, every descriptor newly proposed for an ML model has required a cumbersome and mathematically tedious remodeling. We therefore propose using modern techniques from algorithmic differentiation within the ML modeling process -- effectively enabling the usage of novel descriptors or models fully automatically at an order of magnitude higher computational efficiency. This paradigmatic approach enables not only a versatile usage of novel representations and the efficient computation of larger systems -- all of high value to the FF community -- but also the simple inclusion of further physical knowledge such as higher-order information (e.g. Hessians, more complex partial differential equations constraints etc.), even beyond the presented FF domain.
Lukas Muttenthaler, Robert A. Vandermeulen, Qiuyi Zhang et al.
Model overconfidence and poor calibration are common in machine learning and difficult to account for when applying standard empirical risk minimization. In this work, we propose a novel method to alleviate these problems that we call odd-$k$-out learning (OKO), which minimizes the cross-entropy error for sets rather than for single examples. This naturally allows the model to capture correlations across data examples and achieves both better accuracy and calibration, especially in limited training data and class-imbalanced regimes. Perhaps surprisingly, OKO often yields better calibration even when training with hard labels and dropping any additional calibration parameter tuning, such as temperature scaling. We demonstrate this in extensive experimental analyses and provide a mathematical theory to interpret our findings. We emphasize that OKO is a general framework that can be easily adapted to many settings and a trained model can be applied to single examples at inference time, without significant run-time overhead or architecture changes.
Kirill Bykov, Klaus-Robert Müller, Marina M. -C. Höhne
The utilization of pre-trained networks, especially those trained on ImageNet, has become a common practice in Computer Vision. However, prior research has indicated that a significant number of images in the ImageNet dataset contain watermarks, making pre-trained networks susceptible to learning artifacts such as watermark patterns within their latent spaces. In this paper, we aim to assess the extent to which popular pre-trained architectures display such behavior and to determine which classes are most affected. Additionally, we examine the impact of watermarks on the extracted features. Contrary to the popular belief that the Chinese logographic watermarks impact the "carton" class only, our analysis reveals that a variety of ImageNet classes, such as "monitor", "broom", "apron" and "safe" rely on spurious correlations. Finally, we propose a simple approach to mitigate this issue in fine-tuned networks by ignoring the encodings from the feature-extractor layer of ImageNet pre-trained networks that are most susceptible to watermark imprints.
J. Thorben Frank, Oliver T. Unke, Klaus-Robert Müller et al.
Recent years have seen vast progress in the development of machine learned force fields (MLFFs) based on ab-initio reference calculations. Despite achieving low test errors, the reliability of MLFFs in molecular dynamics (MD) simulations is facing growing scrutiny due to concerns about instability over extended simulation timescales. Our findings suggest a potential connection between robustness to cumulative inaccuracies and the use of equivariant representations in MLFFs, but the computational cost associated with these representations can limit this advantage in practice. To address this, we propose a transformer architecture called SO3krates that combines sparse equivariant representations (Euclidean variables) with a self-attention mechanism that separates invariant and equivariant information, eliminating the need for expensive tensor products. SO3krates achieves a unique combination of accuracy, stability, and speed that enables insightful analysis of quantum properties of matter on extended time and system size scales. To showcase this capability, we generate stable MD trajectories for flexible peptides and supra-molecular structures with hundreds of atoms. Furthermore, we investigate the PES topology for medium-sized chainlike molecules (e.g., small peptides) by exploring thousands of minima. Remarkably, SO3krates demonstrates the ability to strike a balance between the conflicting demands of stability and the emergence of new minimum-energy conformations beyond the training data, which is crucial for realistic exploration tasks in the field of biochemistry.
Stefan Blücher, Klaus-Robert Müller, Stefan Chmiela
Kernel machines have sustained continuous progress in the field of quantum chemistry. In particular, they have proven to be successful in the low-data regime of force field reconstruction. This is because many equivariances and invariances due to physical symmetries can be incorporated into the kernel function to compensate for much larger datasets. So far, the scalability of kernel machines has however been hindered by its quadratic memory and cubical runtime complexity in the number of training points. While it is known, that iterative Krylov subspace solvers can overcome these burdens, their convergence crucially relies on effective preconditioners, which are elusive in practice. Effective preconditioners need to partially pre-solve the learning problem in a computationally cheap and numerically robust manner. Here, we consider the broad class of Nyström-type methods to construct preconditioners based on successively more sophisticated low-rank approximations of the original kernel matrix, each of which provides a different set of computational trade-offs. All considered methods aim to identify a representative subset of inducing (kernel) columns to approximate the dominant kernel spectrum.
Wind turbine power curve models translate ambient conditions into turbine power output. They are essential for energy yield prediction and turbine performance monitoring. In recent years, increasingly complex machine learning methods have become state-of-the-art for this task. Nevertheless, they frequently encounter criticism due to their apparent lack of transparency, which raises concerns regarding their performance in non-stationary environments, such as those faced by wind turbines. We, therefore, introduce an explainable artificial intelligence (XAI) framework to investigate and validate strategies learned by data-driven power curve models from operational wind turbine data. With the help of simple, physics-informed baseline models it enables an automated evaluation of machine learning models beyond standard error metrics. Alongside this novel tool, we present its efficacy for a more informed model selection. We show, for instance, that learned strategies can be meaningful indicators for a model's generalization ability in addition to test set errors, especially when only little data is available. Moreover, the approach facilitates an understanding of how decisions along the machine learning pipeline, such as data selection, pre-processing, or training parameters, affect learned strategies. In a practical example, we demonstrate the framework's utilisation to obtain more physically meaningful models, a prerequisite not only for robustness but also for insights into turbine operation by domain experts. The latter, we demonstrate in the context of wind turbine performance monitoring. Alongside this paper, we publish a Python implementation of the presented framework and hope this can guide researchers and practitioners alike toward training, selecting and utilizing more transparent and robust data-driven wind turbine power curve models.
Hartmut Maennel, Oliver T. Unke, Klaus-Robert Müller
When modeling physical properties of molecules with machine learning, it is desirable to incorporate $SO(3)$-covariance. While such models based on low body order features are not complete, we formulate and prove general completeness properties for higher order methods, and show that $6k-5$ of these features are enough for up to $k$ atoms. We also find that the Clebsch--Gordan operations commonly used in these methods can be replaced by matrix multiplications without sacrificing completeness, lowering the scaling from $O(l^6)$ to $O(l^3)$ in the degree of the features. We apply this to quantum chemistry, but the proposed methods are generally applicable for problems involving 3D point configurations.
Parastoo Semnani, Mihail Bogojeski, Florian Bley et al.
The successful application of machine learning (ML) in catalyst design relies on high-quality and diverse data to ensure effective generalization to novel compositions, thereby aiding in catalyst discovery. However, due to complex interactions, catalyst design has long relied on trial-and-error, a costly and labor-intensive process leading to scarce data that is heavily biased towards undesired, low-yield catalysts. Despite the rise of ML in this field, most efforts have not focused on dealing with the challenges presented by such experimental data. To address these challenges, we introduce a robust machine learning and explainable AI (XAI) framework to accurately classify the catalytic yield of various compositions and identify the contributions of individual components. This framework combines a series of ML practices designed to handle the scarcity and imbalance of catalyst data. We apply the framework to classify the yield of various catalyst compositions in oxidative methane coupling, and use it to evaluate the performance of a range of ML models: tree-based models, logistic regression, support vector machines, and neural networks. These experiments demonstrate that the methods used in our framework lead to a significant improvement in the performance of all but one of the evaluated models. Additionally, the decision-making process of each ML model is analyzed by identifying the most important features for predicting catalyst performance using XAI methods. Our analysis found that XAI methods, providing class-aware explanations, such as Layer-wise Relevance Propagation, identified key components that contribute specifically to high-yield catalysts. These findings align with chemical intuition and existing literature, reinforcing their validity. We believe that such insights can assist chemists in the development and identification of novel catalysts with superior performance.
J. Thorben Frank, Winfried Ripken, Gregor Lied et al. · deepmind
Diffusion Transformers (DiTs) have demonstrated strong performance in generative modeling, particularly in image synthesis, making them a compelling choice for molecular conformer generation. However, applying DiTs to molecules introduces novel challenges, such as integrating discrete molecular graph information with continuous 3D geometry, handling Euclidean symmetries, and designing conditioning mechanisms that generalize across molecules of varying sizes and structures. We propose DiTMC, a framework that adapts DiTs to address these challenges through a modular architecture that separates the processing of 3D coordinates from conditioning on atomic connectivity. To this end, we introduce two complementary graph-based conditioning strategies that integrate seamlessly with the DiT architecture. These are combined with different attention mechanisms, including both standard non-equivariant and SO(3)-equivariant formulations, enabling flexible control over the trade-off between between accuracy and computational efficiency. Experiments on standard conformer generation benchmarks (GEOM-QM9, -DRUGS, -XL) demonstrate that DiTMC achieves state-of-the-art precision and physical validity. Our results highlight how architectural choices and symmetry priors affect sample quality and efficiency, suggesting promising directions for large-scale generative modeling of molecular structures. Code is available at https://github.com/ML4MolSim/dit_mc.
Julius Hense, Mina Jamshidi Idaji, Oliver Eberle et al.
Multiple instance learning (MIL) is an effective and widely used approach for weakly supervised machine learning. In histopathology, MIL models have achieved remarkable success in tasks like tumor detection, biomarker prediction, and outcome prognostication. However, MIL explanation methods are still lagging behind, as they are limited to small bag sizes or disregard instance interactions. We revisit MIL through the lens of explainable AI (XAI) and introduce xMIL, a refined framework with more general assumptions. We demonstrate how to obtain improved MIL explanations using layer-wise relevance propagation (LRP) and conduct extensive evaluation experiments on three toy settings and four real-world histopathology datasets. Our approach consistently outperforms previous explanation attempts with particularly improved faithfulness scores on challenging biomarker prediction tasks. Finally, we showcase how xMIL explanations enable pathologists to extract insights from MIL models, representing a significant advance for knowledge discovery and model debugging in digital histopathology. Codes are available at: https://github.com/bifold-pathomics/xMIL.
Sören Becker, Johanna Vielhaben, Marcel Ackermann et al.
Explainable Artificial Intelligence (XAI) is targeted at understanding how models perform feature selection and derive their classification decisions. This paper explores post-hoc explanations for deep neural networks in the audio domain. Notably, we present a novel Open Source audio dataset consisting of 30,000 audio samples of English spoken digits which we use for classification tasks on spoken digits and speakers' biological sex. We use the popular XAI technique Layer-wise Relevance Propagation (LRP) to identify relevant features for two neural network architectures that process either waveform or spectrogram representations of the data. Based on the relevance scores obtained from LRP, hypotheses about the neural networks' feature selection are derived and subsequently tested through systematic manipulations of the input data. Further, we take a step beyond visual explanations and introduce audible heatmaps. We demonstrate the superior interpretability of audible explanations over visual ones in a human user study.
Jonas Dippel, Barbara Feulner, Tobias Winterhoff et al.
Artificial intelligence has started to transform histopathology impacting clinical diagnostics and biomedical research. However, while many computational pathology approaches have been proposed, most current AI models are limited with respect to generalization, application variety, and handling rare diseases. Recent efforts introduced self-supervised foundation models to address these challenges, yet existing approaches do not leverage pathologist knowledge by design. In this study, we present a novel approach to designing foundation models for computational pathology, incorporating pathologist expertise, semi-automated data curation, and a diverse dataset from over 15 laboratories, including 58 tissue types, and encompassing 129 different histochemical and immunohistochemical staining modalities. We demonstrate that our model "RudolfV" surpasses existing state-of-the-art foundation models across different benchmarks focused on tumor microenvironment profiling, biomarker evaluation, and reference case search while exhibiting favorable robustness properties. Our study shows how domain-specific knowledge can increase the efficiency and performance of pathology foundation models and enable novel application areas.
J. Thorben Frank, Stefan Chmiela, Klaus-Robert Müller et al.
Long-range correlations are essential across numerous machine learning tasks, especially for data embedded in Euclidean space, where the relative positions and orientations of distant components are often critical for accurate predictions. Self-attention offers a compelling mechanism for capturing these global effects, but its quadratic complexity presents a significant practical limitation. This problem is particularly pronounced in computational chemistry, where the stringent efficiency requirements of machine learning force fields (MLFFs) often preclude accurately modeling long-range interactions. To address this, we introduce Euclidean fast attention (EFA), a linear-scaling attention-like mechanism designed for Euclidean data, which can be easily incorporated into existing model architectures. A core component of EFA are novel Euclidean rotary positional encodings (ERoPE), which enable efficient encoding of spatial information while respecting essential physical symmetries. We empirically demonstrate that EFA effectively captures diverse long-range effects, enabling EFA-equipped MLFFs to describe challenging chemical interactions for which conventional MLFFs yield incorrect results.
Maximilian Alber, Stephan Tietz, Jonas Dippel et al.
Recent advances in digital pathology have demonstrated the effectiveness of foundation models across diverse applications. In this report, we present Atlas, a novel vision foundation model based on the RudolfV approach. Our model was trained on a dataset comprising 1.2 million histopathology whole slide images, collected from two medical institutions: Mayo Clinic and Charité - Universtätsmedizin Berlin. Comprehensive evaluations show that Atlas achieves state-of-the-art performance across twenty-one public benchmark datasets, even though it is neither the largest model by parameter count nor by training dataset size.
Dilyara Bareeva, Marina M. -C. Höhne, Alexander Warnecke et al.
Feature Visualization (FV) is a widely used technique for interpreting the concepts learned by Deep Neural Networks (DNNs), which synthesizes input patterns that maximally activate a given feature. Despite its popularity, the trustworthiness of FV explanations has received limited attention. In this paper, we introduce a novel method, Gradient Slingshots, that enables manipulation of FV without modifying the model architecture or significantly degrading its performance. By shaping new trajectories in the off-distribution regions of the activation landscape of a feature, we coerce the optimization process to converge in a predefined visualization. We evaluate our approach on several DNN architectures, demonstrating its ability to replace faithfuls FV with arbitrary targets. These results expose a critical vulnerability: auditors relying solely on FV may accept entirely fabricated explanations. To mitigate this risk, we propose a straightforward defense and quantitatively demonstrate its effectiveness.
Khaled Kahouli, Stefaan Simon Pierre Hessmann, Klaus-Robert Müller et al.
Molecular relaxation, finding the equilibrium state of a non-equilibrium structure, is an essential component of computational chemistry to understand reactivity. Classical force field (FF) methods often rely on insufficient local energy minimization, while neural network FF models require large labeled datasets encompassing both equilibrium and non-equilibrium structures. As a remedy, we propose MoreRed, molecular relaxation by reverse diffusion, a conceptually novel and purely statistical approach where non-equilibrium structures are treated as noisy instances of their corresponding equilibrium states. To enable the denoising of arbitrarily noisy inputs via a generative diffusion model, we further introduce a novel diffusion time step predictor. Notably, MoreRed learns a simpler pseudo potential energy surface (PES) instead of the complex physical PES. It is trained on a significantly smaller, and thus computationally cheaper, dataset consisting of solely unlabeled equilibrium structures, avoiding the computation of non-equilibrium structures altogether. We compare MoreRed to classical FFs, equivariant neural network FFs trained on a large dataset of equilibrium and non-equilibrium data, as well as a semi-empirical tight-binding model. To assess this quantitatively, we evaluate the root-mean-square deviation between the found equilibrium structures and the reference equilibrium structures as well as their energies.
Most current neural networks for molecular dynamics (MD) include physical inductive biases, resulting in specialized and complex architectures. This is in contrast to most other machine learning domains, where specialist approaches are increasingly replaced by general-purpose architectures trained on vast datasets. In line with this trend, several recent studies have questioned the necessity of architectural features commonly found in MD models, such as built-in rotational equivariance or energy conservation. In this work, we contribute to the ongoing discussion by evaluating the performance of an MD model with as few specialized architectural features as possible. We present a recipe for MD using an Edge Transformer, an "off-the-shelf'' transformer architecture that has been minimally modified for the MD domain, termed MD-ET. Our model implements neither built-in equivariance nor energy conservation. We use a simple supervised pre-training scheme on $\sim$30 million molecular structures from the QCML database. Using this "off-the-shelf'' approach, we show state-of-the-art results on several benchmarks after fine-tuning for a small number of steps. Additionally, we examine the effects of being only approximately equivariant and energy conserving for MD simulations, proposing a novel method for distinguishing the errors resulting from non-equivariance from other sources of inaccuracies like numerical rounding errors. While our model exhibits runaway energy increases on larger structures, we show approximately energy-conserving NVE simulations for a range of small structures.
Marco Morik, Ali Hashemi, Klaus-Robert Müller et al.
Reconstructing brain sources is a fundamental challenge in neuroscience, crucial for understanding brain function and dysfunction. Electroencephalography (EEG) signals have a high temporal resolution. However, identifying the correct spatial location of brain sources from these signals remains difficult due to the ill-posed structure of the problem. Traditional methods predominantly rely on manually crafted priors, missing the flexibility of data-driven learning, while recent deep learning approaches focus on end-to-end learning, typically using the physical information of the forward model only for generating training data. We propose the novel hybrid method 3D-PIUNet for EEG source localization that effectively integrates the strengths of traditional and deep learning techniques. 3D-PIUNet starts from an initial physics-informed estimate by using the pseudo inverse to map from measurements to source space. Secondly, by viewing the brain as a 3D volume, we use a 3D convolutional U-Net to capture spatial dependencies and refine the solution according to the learned data prior. Training the model relies on simulated pseudo-realistic brain source data, covering different source distributions. Trained on this data, our model significantly improves spatial accuracy, demonstrating superior performance over both traditional and end-to-end data-driven methods. Additionally, we validate our findings with real EEG data from a visual task, where 3D-PIUNet successfully identifies the visual cortex and reconstructs the expected temporal behavior, thereby showcasing its practical applicability.
Jonah Kömen, Edwin D. de Jong, Julius Hense et al.
Biomedical Foundation Models (FMs) are rapidly transforming AI-enabled healthcare research and entering clinical validation. However, their susceptibility to learning non-biological technical features -- including variations in surgical/endoscopic techniques, laboratory procedures, and scanner hardware -- poses risks for clinical deployment. We present the first systematic investigation of pathology FM robustness to non-biological features. Our work (i) introduces measures to quantify FM robustness, (ii) demonstrates the consequences of limited robustness, and (iii) proposes a framework for FM robustification to mitigate these issues. Specifically, we developed PathoROB, a robustness benchmark with three novel metrics, including the robustness index, and four datasets covering 28 biological classes from 34 medical centers. Our experiments reveal robustness deficits across all 20 evaluated FMs, and substantial robustness differences between them. We found that non-robust FM representations can cause major diagnostic downstream errors and clinical blunders that prevent safe clinical adoption. Using more robust FMs and post-hoc robustification considerably reduced (but did not yet eliminate) the risk of such errors. This work establishes that robustness evaluation is essential for validating pathology FMs before clinical adoption and demonstrates that future FM development must integrate robustness as a core design principle. PathoROB provides a blueprint for assessing robustness across biomedical domains, guiding FM improvement efforts towards more robust, representative, and clinically deployable AI systems that prioritize biological information over technical artifacts.
Sidney Bender, Jan Herrmann, Klaus-Robert Müller et al.
Visual counterfactual explainers (VCEs) are a straightforward and promising approach to enhancing the transparency of image classifiers. VCEs complement other types of explanations, such as feature attribution, by revealing the specific data transformations to which a machine learning model responds most strongly. In this paper, we argue that existing VCEs focus too narrowly on optimizing sample quality or change minimality; they fail to consider the more holistic desiderata for an explanation, such as fidelity, understandability, and sufficiency. To address this shortcoming, we explore new mechanisms for counterfactual generation and investigate how they can help fulfill these desiderata. We combine these mechanisms into a novel 'smooth counterfactual explorer' (SCE) algorithm and demonstrate its effectiveness through systematic evaluations on synthetic and real data.
Simon Letzgus, Klaus-Robert Müller, Grégoire Montavon
In recent years, Explainable AI (XAI) methods have facilitated profound validation and knowledge extraction from ML models. While extensively studied for classification, few XAI solutions have addressed the challenges specific to regression models. In regression, explanations need to be precisely formulated to address specific user queries (e.g.\ distinguishing between `Why is the output above 0?' and `Why is the output above 50?'). They should furthermore reflect the model's behavior on the relevant data sub-manifold. In this paper, we introduce XpertAI, a framework that disentangles the prediction strategy into multiple range-specific sub-strategies and allows the formulation of precise queries about the model (the `explanandum') as a linear combination of those sub-strategies. XpertAI is formulated generally to work alongside popular XAI attribution techniques, based on occlusion, gradient integration, or reverse propagation. Qualitative and quantitative results, demonstrate the benefits of our approach.
Khaled Kahouli, Winfried Ripken, Stefan Gugler et al.
The long sampling time of diffusion models remains a significant bottleneck, which can be mitigated by reducing the number of diffusion time steps. However, the quality of samples with fewer steps is highly dependent on the noise schedule, i.e., the specific manner in which noise is introduced and the signal is reduced at each step. Although prior work has improved upon the original variance-preserving and variance-exploding schedules, these approaches $\textit{passively}$ adjust the total variance, without direct control over it. In this work, we propose a novel total-variance/signal-to-noise-ratio disentangled (TV/SNR) framework, where TV and SNR can be controlled independently. Our approach reveals that schedules where the TV explodes exponentially can often be improved by adopting a constant TV schedule while preserving the same SNR schedule. Furthermore, generalizing the SNR schedule of the optimal transport flow matching significantly improves the generation performance. Our findings hold across various reverse diffusion solvers and diverse applications, including molecular structure and image generation.
While previous studies have demonstrated the potential of AI to diagnose diseases in imaging data, clinical implementation is still lagging behind. This is partly because AI models require training with large numbers of examples only available for common diseases. In clinical reality, however, only few diseases are common, whereas the majority of diseases are less frequent (long-tail distribution). Current AI models overlook or misclassify these diseases. We propose a deep anomaly detection approach that only requires training data from common diseases to detect also all less frequent diseases. We collected two large real-world datasets of gastrointestinal biopsies, which are prototypical of the problem. Herein, the ten most common findings account for approximately 90% of cases, whereas the remaining 10% contained 56 disease entities, including many cancers. 17 million histological images from 5,423 cases were used for training and evaluation. Without any specific training for the diseases, our best-performing model reliably detected a broad spectrum of infrequent ("anomalous") pathologies with 95.0% (stomach) and 91.0% (colon) AUROC and generalized across scanners and hospitals. By design, the proposed anomaly detection can be expected to detect any pathological alteration in the diagnostic tail of gastrointestinal biopsies, including rare primary or metastatic cancers. This study establishes the first effective clinical application of AI-based anomaly detection in histopathology that can flag anomalous cases, facilitate case prioritization, reduce missed diagnoses and enhance the general safety of AI models, thereby driving AI adoption and automation in routine diagnostics and beyond.
Farnoush Rezaei Jafari, Grégoire Montavon, Klaus-Robert Müller et al.
Recent sequence modeling approaches using selective state space sequence models, referred to as Mamba models, have seen a surge of interest. These models allow efficient processing of long sequences in linear time and are rapidly being adopted in a wide range of applications such as language modeling, demonstrating promising performance. To foster their reliable use in real-world scenarios, it is crucial to augment their transparency. Our work bridges this critical gap by bringing explainability, particularly Layer-wise Relevance Propagation (LRP), to the Mamba architecture. Guided by the axiom of relevance conservation, we identify specific components in the Mamba architecture, which cause unfaithful explanations. To remedy this issue, we propose MambaLRP, a novel algorithm within the LRP framework, which ensures a more stable and reliable relevance propagation through these components. Our proposed method is theoretically sound and excels in achieving state-of-the-art explanation performance across a diverse range of models and datasets. Moreover, MambaLRP facilitates a deeper inspection of Mamba architectures, uncovering various biases and evaluating their significance. It also enables the analysis of previous speculations regarding the long-range capabilities of Mamba models.
Kim A. Nicoli, Christopher J. Anders, Lena Funcke et al.
In this paper, we propose a novel and powerful method to harness Bayesian optimization for Variational Quantum Eigensolvers (VQEs) -- a hybrid quantum-classical protocol used to approximate the ground state of a quantum Hamiltonian. Specifically, we derive a VQE-kernel which incorporates important prior information about quantum circuits: the kernel feature map of the VQE-kernel exactly matches the known functional form of the VQE's objective function and thereby significantly reduces the posterior uncertainty. Moreover, we propose a novel acquisition function for Bayesian optimization called Expected Maximum Improvement over Confident Regions (EMICoRe) which can actively exploit the inductive bias of the VQE-kernel by treating regions with low predictive uncertainty as indirectly ``observed''. As a result, observations at as few as three points in the search domain are sufficient to determine the complete objective function along an entire one-dimensional subspace of the optimization landscape. Our numerical experiments demonstrate that our approach improves over state-of-the-art baselines.
Jonas Lederer, Michael Gastegger, Kristof T. Schütt et al.
In recent years, the prediction of quantum mechanical observables with machine learning methods has become increasingly popular. Message-passing neural networks (MPNNs) solve this task by constructing atomic representations, from which the properties of interest are predicted. Here, we introduce a method to automatically identify chemical moieties (molecular building blocks) from such representations, enabling a variety of applications beyond property prediction, which otherwise rely on expert knowledge. The required representation can either be provided by a pretrained MPNN, or learned from scratch using only structural information. Beyond the data-driven design of molecular fingerprints, the versatility of our approach is demonstrated by enabling the selection of representative entries in chemical databases, the automatic construction of coarse-grained force fields, as well as the identification of reaction coordinates.
Transformers have become an important workhorse of machine learning, with numerous applications. This necessitates the development of reliable methods for increasing their transparency. Multiple interpretability methods, often based on gradient information, have been proposed. We show that the gradient in a Transformer reflects the function only locally, and thus fails to reliably identify the contribution of input features to the prediction. We identify Attention Heads and LayerNorm as main reasons for such unreliable explanations and propose a more stable way for propagation through these layers. Our proposal, which can be seen as a proper extension of the well-established LRP method to Transformers, is shown both theoretically and empirically to overcome the deficiency of a simple gradient-based approach, and achieves state-of-the-art explanation performance on a broad range of Transformer models and datasets.
Ann-Kathrin Dombrowski, Klaus-Robert Müller, Wolf Christian Müller
The application of machine learning (ML) techniques, especially neural networks, has seen tremendous success at processing images and language. This is because we often lack formal models to understand visual and audio input, so here neural networks can unfold their abilities as they can model solely from data. In the field of physics we typically have models that describe natural processes reasonably well on a formal level. Nonetheless, in recent years, ML has also proven useful in these realms, be it by speeding up numerical simulations or by improving accuracy. One important and so far unsolved problem in classical physics is understanding turbulent fluid motion. In this work we construct a strongly simplified representation of turbulence by using the Gledzer-Ohkitani-Yamada (GOY) shell model. With this system we intend to investigate the potential of ML-supported and physics-constrained small-scale turbulence modelling. Instead of standard supervised learning we propose an approach that aims to reconstruct statistical properties of turbulence such as the self-similar inertial-range scaling, where we could achieve encouraging experimental results. Furthermore we discuss pitfalls when combining machine learning with differential equations.
Simon Letzgus, Patrick Wagner, Jonas Lederer et al.
In addition to the impressive predictive power of machine learning (ML) models, more recently, explanation methods have emerged that enable an interpretation of complex non-linear learning models such as deep neural networks. Gaining a better understanding is especially important e.g. for safety-critical ML applications or medical diagnostics etc. While such Explainable AI (XAI) techniques have reached significant popularity for classifiers, so far little attention has been devoted to XAI for regression models (XAIR). In this review, we clarify the fundamental conceptual differences of XAI for regression and classification tasks, establish novel theoretical insights and analysis for XAIR, provide demonstrations of XAIR on genuine practical regression problems, and finally discuss the challenges remaining for the field.
Rick Wilming, Céline Budding, Klaus-Robert Müller et al.
Machine learning (ML) is increasingly often used to inform high-stakes decisions. As complex ML models (e.g., deep neural networks) are often considered black boxes, a wealth of procedures has been developed to shed light on their inner workings and the ways in which their predictions come about, defining the field of 'explainable AI' (XAI). Saliency methods rank input features according to some measure of 'importance'. Such methods are difficult to validate since a formal definition of feature importance is, thus far, lacking. It has been demonstrated that some saliency methods can highlight features that have no statistical association with the prediction target (suppressor variables). To avoid misinterpretations due to such behavior, we propose the actual presence of such an association as a necessary condition and objective preliminary definition for feature importance. We carefully crafted a ground-truth dataset in which all statistical dependencies are well-defined and linear, serving as a benchmark to study the problem of suppressor variables. We evaluate common explanation methods including LRP, DTD, PatternNet, PatternAttribution, LIME, Anchors, SHAP, and permutation-based methods with respect to our objective definition. We show that most of these methods are unable to distinguish important features from suppressors in this setting.
Several problems in neuroimaging and beyond require inference on the parameters of multi-task sparse hierarchical regression models. Examples include M/EEG inverse problems, neural encoding models for task-based fMRI analyses, and climate science. In these domains, both the model parameters to be inferred and the measurement noise may exhibit a complex spatio-temporal structure. Existing work either neglects the temporal structure or leads to computationally demanding inference schemes. Overcoming these limitations, we devise a novel flexible hierarchical Bayesian framework within which the spatio-temporal dynamics of model parameters and noise are modeled to have Kronecker product covariance structure. Inference in our framework is based on majorization-minimization optimization and has guaranteed convergence properties. Our highly efficient algorithms exploit the intrinsic Riemannian geometry of temporal autocovariance matrices. For stationary dynamics described by Toeplitz matrices, the theory of circulant embeddings is employed. We prove convex bounding properties and derive update rules of the resulting algorithms. On both synthetic and real neural data from M/EEG, we demonstrate that our methods lead to improved performance.
Armin W. Thomas, Ulman Lindenberger, Wojciech Samek et al.
Research in many fields has shown that transfer learning (TL) is well-suited to improve the performance of deep learning (DL) models in datasets with small numbers of samples. This empirical success has triggered interest in the application of TL to cognitive decoding analyses with functional neuroimaging data. Here, we systematically evaluate TL for the application of DL models to the decoding of cognitive states (e.g., viewing images of faces or houses) from whole-brain functional Magnetic Resonance Imaging (fMRI) data. We first pre-train two DL architectures on a large, public fMRI dataset and subsequently evaluate their performance in an independent experimental task and a fully independent dataset. The pre-trained models consistently achieve higher decoding accuracies and generally require less training time and data than model variants that were not pre-trained, clearly underlining the benefits of pre-training. We demonstrate that these benefits arise from the ability of the pre-trained models to reuse many of their learned features when training with new data, providing deeper insights into the mechanisms giving rise to the benefits of pre-training. Yet, we also surface nuanced challenges for whole-brain cognitive decoding with DL models when interpreting the decoding decisions of the pre-trained models, as these have learned to utilize the fMRI data in unforeseen and counterintuitive ways to identify individual cognitive states.
Niklas W. A. Gebauer, Michael Gastegger, Stefaan S. P. Hessmann et al.
The rational design of molecules with desired properties is a long-standing challenge in chemistry. Generative neural networks have emerged as a powerful approach to sample novel molecules from a learned distribution. Here, we propose a conditional generative neural network for 3d molecular structures with specified chemical and structural properties. This approach is agnostic to chemical bonding and enables targeted sampling of novel molecules from conditional distributions, even in domains where reference calculations are sparse. We demonstrate the utility of our method for inverse design by generating molecules with specified motifs or composition, discovering particularly stable molecules, and jointly targeting multiple electronic properties beyond the training regime.
Kirill Bykov, Marina M. -C. Höhne, Adelaida Creosteanu et al.
To advance the transparency of learning machines such as Deep Neural Networks (DNNs), the field of Explainable AI (XAI) was established to provide interpretations of DNNs' predictions. While different explanation techniques exist, a popular approach is given in the form of attribution maps, which illustrate, given a particular data point, the relevant patterns the model has used for making its prediction. Although Bayesian models such as Bayesian Neural Networks (BNNs) have a limited form of transparency built-in through their prior weight distribution, they lack explanations of their predictions for given instances. In this work, we take a step toward combining these two perspectives by examining how local attributions can be extended to BNNs. Within the Bayesian framework, network weights follow a probability distribution; hence, the standard point explanation extends naturally to an explanation distribution. Viewing explanations probabilistically, we aggregate and analyze multiple local attributions drawn from an approximate posterior to explore variability in explanation patterns. The diversity of explanations offers a way to further explore how predictive rationales may vary across posterior samples. Quantitative and qualitative experiments on toy and benchmark data, as well as on a real-world pathology dataset, illustrate that our framework enriches standard explanations with uncertainty information and may support the visualization of explanation stability.
Vignesh Srinivasan, Nils Strodthoff, Jackie Ma et al.
There is an increasing number of medical use-cases where classification algorithms based on deep neural networks reach performance levels that are competitive with human medical experts. To alleviate the challenges of small dataset sizes, these systems often rely on pretraining. In this work, we aim to assess the broader implications of these approaches. For diabetic retinopathy grading as exemplary use case, we compare the impact of different training procedures including recently established self-supervised pretraining methods based on contrastive learning. To this end, we investigate different aspects such as quantitative performance, statistics of the learned feature representations, interpretability and robustness to image distortions. Our results indicate that models initialized from ImageNet pretraining report a significant increase in performance, generalization and robustness to image distortions. In particular, self-supervised models show further benefits to supervised models. Self-supervised models with initialization from ImageNet pretraining not only report higher performance, they also reduce overfitting to large lesions along with improvements in taking into account minute lesions indicative of the progression of the disease. Understanding the effects of pretraining in a broader sense that goes beyond simple performance comparisons is of crucial importance for the broader medical imaging community beyond the use-case considered in this work.
Christopher J. Anders, David Neumann, Wojciech Samek et al.
Deep Neural Networks (DNNs) are known to be strong predictors, but their prediction strategies can rarely be understood. With recent advances in Explainable Artificial Intelligence (XAI), approaches are available to explore the reasoning behind those complex models' predictions. Among post-hoc attribution methods, Layer-wise Relevance Propagation (LRP) shows high performance. For deeper quantitative analysis, manual approaches exist, but without the right tools they are unnecessarily labor intensive. In this software paper, we introduce three software packages targeted at scientists to explore model reasoning using attribution approaches and beyond: (1) Zennit - a highly customizable and intuitive attribution framework implementing LRP and related approaches in PyTorch, (2) CoRelAy - a framework to easily and quickly construct quantitative analysis pipelines for dataset-wide analyses of explanations, and (3) ViRelAy - a web-application to interactively explore data, attributions, and analysis results. With this, we provide a standardized implementation solution for XAI, to contribute towards more reproducibility in our field.
Domain shifts in the training data are common in practical applications of machine learning; they occur for instance when the data is coming from different sources. Ideally, a ML model should work well independently of these shifts, for example, by learning a domain-invariant representation. However, common ML losses do not give strong guarantees on how consistently the ML model performs for different domains, in particular, whether the model performs well on a domain at the expense of its performance on another domain. In this paper, we build new theoretical foundations for this problem, by contributing a set of mathematical relations between classical losses for supervised ML and the Wasserstein distance in joint space (i.e. representation and output space). We show that classification or regression losses, when combined with a GAN-type discriminator between domains, form an upper-bound to the true Wasserstein distance between domains. This implies a more invariant representation and also more stable prediction performance across domains. Theoretical results are corroborated empirically on several image datasets. Our proposed approach systematically produces the highest minimum classification accuracy across domains, and the most invariant representation.
Huziel E. Sauceda, Luis E. Gálvez-González, Stefan Chmiela et al.
Machine-learning force fields (MLFF) should be accurate, computationally and data efficient, and applicable to molecules, materials, and interfaces thereof. Currently, MLFFs often introduce tradeoffs that restrict their practical applicability to small subsets of chemical space or require exhaustive datasets for training. Here, we introduce the Bravais-Inspired Gradient-Domain Machine Learning (BIGDML) approach and demonstrate its ability to construct reliable force fields using a training set with just 10-200 geometries for materials including pristine and defect-containing 2D and 3D semiconductors and metals, as well as chemisorbed and physisorbed atomic and molecular adsorbates on surfaces. The BIGDML model employs the full relevant symmetry group for a given material, does not assume artificial atom types or localization of atomic interactions and exhibits high data efficiency and state-of-the-art energy accuracies (errors substantially below 1 meV per atom) for an extended set of materials. Extensive path-integral molecular dynamics carried out with BIGDML models demonstrate the counterintuitive localization of benzene--graphene dynamics induced by nuclear quantum effects and allow to rationalize the Arrhenius behavior of hydrogen diffusion coefficient in a Pd crystal for a wide range of temperatures.