Felix Dietrich

Nikolaos Evangelou, Felix Dietrich, Eliodoro Chiavazzo et al.

We introduce a data-driven approach to building reduced dynamical models through manifold learning; the reduced latent space is discovered using Diffusion Maps (a manifold learning technique) on time series data. A second round of Diffusion Maps on those latent coordinates allows the approximation of the reduced dynamical models. This second round enables mapping the latent space coordinates back to the full ambient space (what is called lifting); it also enables the approximation of full state functions of interest in terms of the reduced coordinates. In our work, we develop and test three different reduced numerical simulation methodologies, either through pre-tabulation in the latent space and integration on the fly or by going back and forth between the ambient space and the latent space. The data-driven latent space simulation results, based on the three different approaches, are validated through (a) the latent space observation of the full simulation through the Nyström Extension formula, or through (b) lifting the reduced trajectory back to the full ambient space, via Latent Harmonics. Latent space modeling often involves additional regularization to favor certain properties of the space over others, and the mapping back to the ambient space is then constructed mostly independently from these properties; here, we use the same data-driven approach to construct the latent space and then map back to the ambient space.

Danimir T. Doncevic, Alexander Mitsos, Yue Guo et al.

Meta-learning of numerical algorithms for a given task consists of the data-driven identification and adaptation of an algorithmic structure and the associated hyperparameters. To limit the complexity of the meta-learning problem, neural architectures with a certain inductive bias towards favorable algorithmic structures can, and should, be used. We generalize our previously introduced Runge-Kutta neural network to a recursively recurrent neural network (R2N2) superstructure for the design of customized iterative algorithms. In contrast to off-the-shelf deep learning approaches, it features a distinct division into modules for generation of information and for the subsequent assembly of this information towards a solution. Local information in the form of a subspace is generated by subordinate, inner, iterations of recurrent function evaluations starting at the current outer iterate. The update to the next outer iterate is computed as a linear combination of these evaluations, reducing the residual in this space, and constitutes the output of the network. We demonstrate that regular training of the weight parameters inside the proposed superstructure on input/output data of various computational problem classes yields iterations similar to Krylov solvers for linear equation systems, Newton-Krylov solvers for nonlinear equation systems, and Runge-Kutta integrators for ordinary differential equations. Due to its modularity, the superstructure can be readily extended with functionalities needed to represent more general classes of iterative algorithms traditionally based on Taylor series expansions.

Ioannis K. Gallos, Daniel Lehmberg, Felix Dietrich et al.

We propose a machine-learning approach to model long-term out-of-sample dynamics of brain activity from task-dependent fMRI data. Our approach is a three stage one. First, we exploit Diffusion maps (DMs) to discover a set of variables that parametrize the low-dimensional manifold on which the emergent high-dimensional fMRI time series evolve. Then, we construct reduced-order-models (ROMs) on the embedded manifold via two techniques: Feedforward Neural Networks (FNNs) and the Koopman operator. Finally, for predicting the out-of-sample long-term dynamics of brain activity in the ambient fMRI space, we solve the pre-image problem coupling DMs with Geometric Harmonics (GH) when using FNNs and the Koopman modes per se. For our illustrations, we have assessed the performance of the two proposed schemes using a benchmark fMRI dataset with recordings during a visuo-motor task. The results suggest that just a few (for the particular task, five) non-linear coordinates of the high-dimensional fMRI time series provide a good basis for modelling and out-of-sample prediction of the brain activity. Furthermore, we show that the proposed approaches outperform the one-step ahead predictions of the naive random walk model, which, in contrast to our scheme, relies on the knowledge of the signals in the previous time step. Importantly, we show that the proposed Koopman operator approach provides, for any practical purposes, equivalent results to the FNN-GH approach, thus bypassing the need to train a non-linear map and to use GH to extrapolate predictions in the ambient fMRI space; one can use instead the low-frequency truncation of the DMs function space of L^2-integrable functions, to predict the entire list of coordinate functions in the fMRI space and to solve the pre-image problem.

Nikolaos Evangelou, Felix Dietrich, Juan M. Bello-Rivas et al.

We construct a reduced, data-driven, parameter dependent effective Stochastic Differential Equation (eSDE) for electric-field mediated colloidal crystallization using data obtained from Brownian Dynamics Simulations. We use Diffusion Maps (a manifold learning algorithm) to identify a set of useful latent observables. In this latent space we identify an eSDE using a deep learning architecture inspired by numerical stochastic integrators and compare it with the traditional Kramers-Moyal expansion estimation. We show that the obtained variables and the learned dynamics accurately encode the physics of the Brownian Dynamic Simulations. We further illustrate that our reduced model captures the dynamics of corresponding experimental data. Our dimension reduction/reduced model identification approach can be easily ported to a broad class of particle systems dynamics experiments/models.

Erik Lien Bolager, Iryna Burak, Chinmay Datar et al.

We introduce a probability distribution, combined with an efficient sampling algorithm, for weights and biases of fully-connected neural networks. In a supervised learning context, no iterative optimization or gradient computations of internal network parameters are needed to obtain a trained network. The sampling is based on the idea of random feature models. However, instead of a data-agnostic distribution, e.g., a normal distribution, we use both the input and the output training data to sample shallow and deep networks. We prove that sampled networks are universal approximators. For Barron functions, we show that the $L^2$-approximation error of sampled shallow networks decreases with the square root of the number of neurons. Our sampling scheme is invariant to rigid body transformations and scaling of the input data, which implies many popular pre-processing techniques are not required. In numerical experiments, we demonstrate that sampled networks achieve accuracy comparable to iteratively trained ones, but can be constructed orders of magnitude faster. Our test cases involve a classification benchmark from OpenML, sampling of neural operators to represent maps in function spaces, and transfer learning using well-known architectures.

Philipp Scholl, Felix Dietrich, Clemens Otte et al.

Safe Policy Improvement (SPI) is an important technique for offline reinforcement learning in safety critical applications as it improves the behavior policy with a high probability. We classify various SPI approaches from the literature into two groups, based on how they utilize the uncertainty of state-action pairs. Focusing on the Soft-SPIBB (Safe Policy Improvement with Soft Baseline Bootstrapping) algorithms, we show that their claim of being provably safe does not hold. Based on this finding, we develop adaptations, the Adv-Soft-SPIBB algorithms, and show that they are provably safe. A heuristic adaptation, Lower-Approx-Soft-SPIBB, yields the best performance among all SPIBB algorithms in extensive experiments on two benchmarks. We also check the safety guarantees of the provably safe algorithms and show that huge amounts of data are necessary such that the safety bounds become useful in practice.

Felix Dietrich, Dennis Merkert, Bernd Simeon

In this paper, we first introduce the reader to the Basic Scheme of Moulinec and Suquet in the setting of quasi-static linear elasticity, which takes advantage of the fast Fourier transform on homogenized microstructures to accelerate otherwise time-consuming computations. By means of an asymptotic expansion, a hierarchy of linear problems is derived, whose solutions are looked at in detail. It is highlighted how these generalized homogenization problems depend on each other. We extend the Basic Scheme to fit this new problem class and give some numerical results for the first two problem orders.

Chinmay Datar, Taniya Kapoor, Abhishek Chandra et al.

Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their accuracy and training speed are limited by two core barriers: gradient-descent-based iterative optimization over complex loss landscapes and non-causal treatment of time as an extra spatial dimension. We present Frozen-PINN, a novel PINN based on the principle of space-time separation that leverages random features instead of training with gradient descent, and incorporates temporal causality by construction. On eight PDE benchmarks, including challenges such as extreme advection speeds, shocks, and high dimensionality, Frozen-PINNs achieve superior training efficiency and accuracy over state-of-the-art PINNs, often by several orders of magnitude. Our work addresses longstanding training and accuracy bottlenecks of PINNs, delivering quickly trainable, highly accurate, and inherently causal PDE solvers, a combination that prior methods could not realize. Our approach challenges the reliance of PINNs on stochastic gradient-descent-based methods and specialized hardware, leading to a paradigm shift in PINN training and providing a challenging benchmark for the community.

Divya Shyam Singh, Leon Herrmann, Qing Sun et al.

Full waveform inversion (FWI) is a powerful tool for reconstructing material fields based on sparsely measured data obtained by wave propagation. For specific problems, discretizing the material field with a neural network (NN) improves the robustness and reconstruction quality of the corresponding optimization problem. We call this method NN-based FWI. Starting from an initial guess, the weights of the NN are iteratively updated to fit the simulated wave signals to the sparsely measured data set. For gradient-based optimization, a suitable choice of the initial guess, i.e., a suitable NN weight initialization, is crucial for fast and robust convergence. In this paper, we introduce a novel transfer learning approach to further improve NN-based FWI. This approach leverages supervised pretraining to provide a better NN weight initialization, leading to faster convergence of the subsequent optimization problem. Moreover, the inversions yield physically more meaningful local minima. The network is pretrained to predict the unknown material field using the gradient information from the first iteration of conventional FWI. In our computational experiments on two-dimensional domains, the training data set consists of reference simulations with arbitrarily positioned elliptical voids of different shapes and orientations. We compare the performance of the proposed transfer learning NN-based FWI with three other methods: conventional FWI, NN-based FWI without pretraining and conventional FWI with an initial guess predicted from the pretrained NN. Our results show that transfer learning NN-based FWI outperforms the other methods in terms of convergence speed and reconstruction quality.

Atamert Rahma, Chinmay Datar, Ana Cukarska et al.

Learning dynamical systems that respect physical symmetries and constraints remains a fundamental challenge in data-driven modeling. Integrating physical laws with graph neural networks facilitates principled modeling of complex N-body dynamics and yields accurate and permutation-invariant models. However, training graph neural networks with iterative, gradient-descent-based optimization algorithms (e.g., Adam, RMSProp, LBFGS) often leads to slow training, especially for large, complex systems. In comparison to 15 different optimizers, we demonstrate that Hamiltonian Graph Networks (HGN) can be trained 150-600x faster - but with comparable accuracy - by replacing iterative optimization with random feature-based parameter construction. We show robust performance in diverse simulations, including N-body mass-spring and molecular dynamics systems in up to dimensions and 10,000 particles with different geometries, while retaining essential physical invariances with respect to permutation, rotation, and translation. Our proposed approach is benchmarked using a NeurIPS 2022 Datasets and Benchmarks Track publication to further demonstrate its versatility. We reveal that even when trained on minimal 8-node systems, the model can generalize in a zero-shot manner to systems as large as 4096 nodes without retraining. Our work challenges the dominance of iterative gradient-descent-based optimization algorithms for training neural network models for physical systems.

Erik Lien Bolager, Boumediene Hamzi, Houman Owhadi et al.

Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD we show how to perform gradient-based optimization over the learnable kernel parameters, and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto-Sivashinsky PDE, showcasing the method's different strengths.

Kislaya Ravi, Vladyslav Fediukov, Felix Dietrich et al.

One of the main challenges in surrogate modeling is the limited availability of data due to resource constraints associated with computationally expensive simulations. Multi-fidelity methods provide a solution by chaining models in a hierarchy with increasing fidelity, associated with lower error, but increasing cost. In this paper, we compare different multi-fidelity methods employed in constructing Gaussian process surrogates for regression. Non-linear autoregressive methods in the existing literature are primarily confined to two-fidelity models, and we extend these methods to handle more than two levels of fidelity. Additionally, we propose enhancements for an existing method incorporating delay terms by introducing a structured kernel. We demonstrate the performance of these methods across various academic and real-world scenarios. Our findings reveal that multi-fidelity methods generally have a smaller prediction error for the same computational cost as compared to the single-fidelity method, although their effectiveness varies across different scenarios.

Erik Lien Bolager, Ana Cukarska, Iryna Burak et al.

Recurrent neural networks are a successful neural architecture for many time-dependent problems, including time series analysis, forecasting, and modeling of dynamical systems. Training such networks with backpropagation through time is a notoriously difficult problem because their loss gradients tend to explode or vanish. In this contribution, we introduce a computational approach to construct all weights and biases of a recurrent neural network without using gradient-based methods. The approach is based on a combination of random feature networks and Koopman operator theory for dynamical systems. The hidden parameters of a single recurrent block are sampled at random, while the outer weights are constructed using extended dynamic mode decomposition. This approach alleviates all problems with backpropagation commonly related to recurrent networks. The connection to Koopman operator theory also allows us to start using results in this area to analyze recurrent neural networks. In computational experiments on time series, forecasting for chaotic dynamical systems, and control problems, as well as on weather data, we observe that the training time and forecasting accuracy of the recurrent neural networks we construct are improved when compared to commonly used gradient-based methods.

Zahra Monfared, Saksham Malhotra, Sekiya Hajime et al.

For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator of the system form a multiplicative group. Here, we exploit this property to accelerate the systematic numerical computation of the eigenspaces of the operator. Given a small set of (so-called ``principal'') eigenfunctions that are approximated conventionally, we can obtain a much larger set by constructing polynomials of the principal eigenfunctions. This enriches the set, and thus allows us to more accurately represent application-specific observables. Often, eigenfunctions exhibit localized singularities (e.g. in simple, one-dimensional problems with multiple steady states) or extended ones (e.g. in simple, two-dimensional problems possessing a limit cycle, or a separatrix); we discuss eigenfunction matching/continuation across such singularities. By handling eigenfunction singularities and enabling their continuation, our approach supports learning consistent global representations from locally sampled data. This is particularly relevant for multistable systems and applications with sparse or fragmented measurements.

Sucheta Ghosh, Zahra Monfared, Felix Dietrich

We introduce a two-stage multitask learning framework for analyzing Electroencephalography (EEG) signals that integrates denoising, dynamical modeling, and representation learning. In the first stage, a denoising autoencoder is trained to suppress artifacts and stabilize temporal dynamics, providing robust signal representations. In the second stage, a multitask architecture processes these denoised signals to achieve three objectives: motor imagery classification, chaotic versus non-chaotic regime discrimination using Lyapunov exponent-based labels, and self-supervised contrastive representation learning with NT-Xent loss. A convolutional backbone combined with a Transformer encoder captures spatial-temporal structure, while the dynamical task encourages sensitivity to nonlinear brain dynamics. This staged design mitigates interference between reconstruction and discriminative goals, improves stability across datasets, and supports reproducible training by clearly separating noise reduction from higher-level feature learning. Empirical studies show that our framework not only enhances robustness and generalization but also surpasses strong baselines and recent state-of-the-art methods in EEG decoding, highlighting the effectiveness of combining denoising, dynamical features, and self-supervised learning.

Atamert Rahma, Chinmay Datar, Felix Dietrich

Neural networks that synergistically integrate data and physical laws offer great promise in modeling dynamical systems. However, iterative gradient-based optimization of network parameters is often computationally expensive and suffers from slow convergence. In this work, we present a backpropagation-free algorithm to accelerate the training of neural networks for approximating Hamiltonian systems through data-agnostic and data-driven algorithms. We empirically show that data-driven sampling of the network parameters outperforms data-agnostic sampling or the traditional gradient-based iterative optimization of the network parameters when approximating functions with steep gradients or wide input domains. We demonstrate that our approach is more than 100 times faster with CPUs than the traditionally trained Hamiltonian Neural Networks using gradient-based iterative optimization and is more than four orders of magnitude accurate in chaotic examples, including the Hénon-Heiles system.

Erez Peterfreund, Iryna Burak, Ofir Lindenbaum et al.

Fusing measurements from multiple, heterogeneous, partial sources, observing a common object or process, poses challenges due to the increasing availability of numbers and types of sensors. In this work we propose, implement and validate an end-to-end computational pipeline in the form of a multiple-auto-encoder neural network architecture for this task. The inputs to the pipeline are several sets of partial observations, and the result is a globally consistent latent space, harmonizing (rigidifying, fusing) all measurements. The key enabler is the availability of multiple slightly perturbed measurements of each instance:, local measurement, "bursts", that allows us to estimate the local distortion induced by each instrument. We demonstrate the approach in a sequence of examples, starting with simple two-dimensional data sets and proceeding to a Wi-Fi localization problem and to the solution of a "dynamical puzzle" arising in spatio-temporal observations of the solutions of Partial Differential Equations.

Chinmay Datar, Adwait Datar, Felix Dietrich et al.

Discovering a suitable neural network architecture for modeling complex dynamical systems poses a formidable challenge, often involving extensive trial and error and navigation through a high-dimensional hyper-parameter space. In this paper, we discuss a systematic approach to constructing neural architectures for modeling a subclass of dynamical systems, namely, Linear Time-Invariant (LTI) systems. We use a variant of continuous-time neural networks in which the output of each neuron evolves continuously as a solution of a first-order or second-order Ordinary Differential Equation (ODE). Instead of deriving the network architecture and parameters from data, we propose a gradient-free algorithm to compute sparse architecture and network parameters directly from the given LTI system, leveraging its properties. We bring forth a novel neural architecture paradigm featuring horizontal hidden layers and provide insights into why employing conventional neural architectures with vertical hidden layers may not be favorable. We also provide an upper bound on the numerical errors of our neural networks. Finally, we demonstrate the high accuracy of our constructed networks on three numerical examples.

Maximilian Gollwitzer, Felix Dietrich

Spiking Neural Networks (SNNs) as Machine Learning (ML) models have recently received a lot of attention as a potentially more energy-efficient alternative to conventional Artificial Neural Networks. The non-differentiability and sparsity of the spiking mechanism can make these models very difficult to train with algorithms based on propagating gradients through the spiking non-linearity. We address this problem by adapting the paradigm of Random Feature Methods (RFMs) from Artificial Neural Networks (ANNs) to Spike Response Model (SRM) SNNs. This approach allows training of SNNs without approximation of the spike function gradient. Concretely, we propose a novel data-driven, fast, high-performance, and interpretable algorithm for end-to-end training of SNNs inspired by the SWIM algorithm for RFM-ANNs, which we coin S-SWIM. We provide a thorough theoretical discussion and supplementary numerical experiments showing that S-SWIM can reach high accuracies on time series forecasting as a standalone strategy and serve as an effective initialisation strategy before gradient-based training. Additional ablation studies show that our proposed method performs better than random sampling of network weights.

Jiajie Fan, Amal Trigui, Andrea Bonfanti et al.

Recent advancements in learning latent codes derived from high-dimensional shapes have demonstrated impressive outcomes in 3D generative modeling. Traditionally, these approaches employ a trained autoencoder to acquire a continuous implicit representation of source shapes, which can be computationally expensive. This paper introduces a novel framework, spectral-domain diffusion for high-quality shape generation SpoDify, that utilizes singular value decomposition (SVD) for shape encoding. The resulting eigenvectors can be stored for subsequent decoding, while generative modeling is performed on the eigenfeatures. This approach efficiently encodes complex meshes into continuous implicit representations, such as encoding a 15k-vertex mesh to a 512-dimensional latent code without learning. Our method exhibits significant advantages in scenarios with limited samples or GPU resources. In mesh generation tasks, our approach produces high-quality shapes that are comparable to state-of-the-art methods.

David W. Sroczynski, Felix Dietrich, Eleni D. Koronaki et al.

Before we attempt to learn a function between two (sets of) observables of a physical process, we must first decide what the inputs and what the outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of initially deciding ``the right quantities'' to relate through such a function, and then proceed to learn it. This is accomplished by processing multiple simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: multiple observation processes of the system. We thus determine (a) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (b) what information is unrelated to these common observables, and therefore particular to each observation process, and not contributing to the desired function. Any data-driven function approximation technique can subsequently be used to learn the input-output relation, from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular ``twists'' of the approach are discussed. The first has to do with the identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations of one process to entire level sets of measurements of the process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures ``now'', while the second observation process measures ``in the future'', the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution.

Philipp Scholl, Felix Dietrich, Clemens Otte et al.

Safe Policy Improvement (SPI) aims at provable guarantees that a learned policy is at least approximately as good as a given baseline policy. Building on SPI with Soft Baseline Bootstrapping (Soft-SPIBB) by Nadjahi et al., we identify theoretical issues in their approach, provide a corrected theory, and derive a new algorithm that is provably safe on finite Markov Decision Processes (MDP). Additionally, we provide a heuristic algorithm that exhibits the best performance among many state of the art SPI algorithms on two different benchmarks. Furthermore, we introduce a taxonomy of SPI algorithms and empirically show an interesting property of two classes of SPI algorithms: while the mean performance of algorithms that incorporate the uncertainty as a penalty on the action-value is higher, actively restricting the set of policies more consistently produces good policies and is, thus, safer.

Nikolaos Evangelou, Noah J. Wichrowski, George A. Kevrekidis et al.

We present a data-driven approach to characterizing nonidentifiability of a model's parameters and illustrate it through dynamic as well as steady kinetic models. By employing Diffusion Maps and their extensions, we discover the minimal combinations of parameters required to characterize the output behavior of a chemical system: a set of effective parameters for the model. Furthermore, we introduce and use a Conformal Autoencoder Neural Network technique, as well as a kernel-based Jointly Smooth Function technique, to disentangle the redundant parameter combinations that do not affect the output behavior from the ones that do. We discuss the interpretability of our data-driven effective parameters, and demonstrate the utility of the approach both for behavior prediction and parameter estimation. In the latter task, it becomes important to describe level sets in parameter space that are consistent with a particular output behavior. We validate our approach on a model of multisite phosphorylation, where a reduced set of effective parameters (nonlinear combinations of the physical ones) has previously been established analytically.

Felix Dietrich, Juan M. Bello-Rivas, Ioannis G. Kevrekidis

We discuss the correspondence between Gaussian process regression and Geometric Harmonics, two similar kernel-based methods that are typically used in different contexts. Research communities surrounding the two concepts often pursue different goals. Results from both camps can be successfully combined, providing alternative interpretations of uncertainty in terms of error estimation, or leading towards accelerated Bayesian Optimization due to dimensionality reduction.

Yubin Lu, Romit Maulik, Ting Gao et al.

In this work, we propose a method to learn multivariate probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning assisted construction of a time-dependent mapping that takes a reference distribution (say, a Gaussian) to each and every instance of our evolving distribution. If the reference distribution is the initial condition of a Fokker-Planck equation, what we learn is the time-T map of the corresponding solution. Specifically, the learned map is a multivariate normalizing flow that deforms the support of the reference density to the support of each and every density snapshot in time. We demonstrate that this approach can approximate probability density function evolutions in time from observed sampled data for systems driven by both Brownian and Lévy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.

Felix Dietrich, Alexei Makeev, George Kevrekidis et al.

We identify effective stochastic differential equations (SDE) for coarse observables of fine-grained particle- or agent-based simulations; these SDE then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDE through neural networks, which can be thought of as effective stochastic ResNets. The loss function is inspired by, and embodies, the structure of established stochastic numerical integrators (here, Euler-Maruyama and Milstein); our approximations can thus benefit from backward error analysis of these underlying numerical schemes. They also lend themselves naturally to "physics-informed" gray-box identification when approximate coarse models, such as mean field equations, are available. Existing numerical integration schemes for Langevin-type equations and for stochastic partial differential equations (SPDE) can also be used for training; we demonstrate this on a stochastically forced oscillator and the stochastic wave equation. Our approach does not require long trajectories, works on scattered snapshot data, and is designed to naturally handle different time steps per snapshot. We consider both the case where the coarse collective observables are known in advance, as well as the case where they must be found in a data-driven manner.

Yue Guo, Felix Dietrich, Tom Bertalan et al.

We study the learning of numerical algorithms for scientific computing, which combines mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a departure from the classical approaches in numerical analysis, which typically do not feature such learning-based adaptations. As a case study, we develop a machine learning approach that automatically learns effective solvers for initial value problems in the form of ordinary differential equations (ODEs), based on the Runge-Kutta (RK) integrator architecture. We show that we can learn high-order integrators for targeted families of differential equations without the need for computing integrator coefficients by hand. Moreover, we demonstrate that in certain cases we can obtain superior performance to classical RK methods. This can be attributed to certain properties of the ODE families being identified and exploited by the approach. Overall, this work demonstrates an effective learning-based approach to the design of algorithms for the numerical solution of differential equations. This can be readily extended to other numerical tasks.

Felix P. Kemeth, Tom Bertalan, Thomas Thiem et al.

We extract data-driven, intrinsic spatial coordinates from observations of the dynamics of large systems of coupled heterogeneous agents. These coordinates then serve as an emergent space in which to learn predictive models in the form of partial differential equations (PDEs) for the collective description of the coupled-agent system. They play the role of the independent spatial variables in this PDE (as opposed to the dependent, possibly also data-driven, state variables). This leads to an alternative description of the dynamics, local in these emergent coordinates, thus facilitating an alternative modeling path for complex coupled-agent systems. We illustrate this approach on a system where each agent is a limit cycle oscillator (a so-called Stuart-Landau oscillator); the agents are heterogeneous (they each have a different intrinsic frequency $ω$) and are coupled through the ensemble average of their respective variables. After fast initial transients, we show that the collective dynamics on a slow manifold can be approximated through a learned model based on local "spatial" partial derivatives in the emergent coordinates. The model is then used for prediction in time, as well as to capture collective bifurcations when system parameters vary. The proposed approach thus integrates the automatic, data-driven extraction of emergent space coordinates parametrizing the agent dynamics, with machine-learning assisted identification of an "emergent PDE" description of the dynamics in this parametrization.

Tom Bertalan, Felix Dietrich, Ioannis G. Kevrekidis

We propose to test, and when possible establish, an equivalence between two different artificial neural networks by attempting to construct a data-driven transformation between them, using manifold-learning techniques. In particular, we employ diffusion maps with a Mahalanobis-like metric. If the construction succeeds, the two networks can be thought of as belonging to the same equivalence class. We first discuss transformation functions between only the outputs of the two networks; we then also consider transformations that take into account outputs (activations) of a number of internal neurons from each network. In general, Whitney's theorem dictates the number of measurements from one of the networks required to reconstruct each and every feature of the second network. The construction of the transformation function relies on a consistent, intrinsic representation of the network input space. We illustrate our algorithm by matching neural network pairs trained to learn (a) observations of scalar functions; (b) observations of two-dimensional vector fields; and (c) representations of images of a moving three-dimensional object (a rotating horse). The construction of such equivalence classes across different network instantiations clearly relates to transfer learning. We also expect that it will be valuable in establishing equivalence between different Machine Learning-based models of the same phenomenon observed through different instruments and by different research groups.

Erez Peterfreund, Ofir Lindenbaum, Felix Dietrich et al.

We propose a deep-learning based method for obtaining standardized data coordinates from scientific measurements.Data observations are modeled as samples from an unknown, non-linear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized latent variables. By leveraging a repeated measurement sampling strategy, we present a method for learning an embedding in $\mathbb{R}^d$ that is isometric to the latent variables of the manifold. These data coordinates, being invariant under smooth changes of variables, enable matching between different instrumental observations of the same phenomenon. Our embedding is obtained using a LOcal Conformal Autoencoder (LOCA), an algorithm that constructs an embedding to rectify deformations by using a local z-scoring procedure while preserving relevant geometric information. We demonstrate the isometric embedding properties of LOCA on various model settings and observe that it exhibits promising interpolation and extrapolation capabilities. Finally, we apply LOCA to single-site Wi-Fi localization data, and to $3$-dimensional curved surface estimation based on a $2$-dimensional projection.

Felix Dietrich, Or Yair, Rotem Mulayoff et al.

In this paper, we propose a spectral method for deriving functions that are jointly smooth on multiple observed manifolds. This allows us to register measurements of the same phenomenon by heterogeneous sensors, and to reject sensor-specific noise. Our method is unsupervised and primarily consists of two steps. First, using kernels, we obtain a subspace spanning smooth functions on each separate manifold. Then, we apply a spectral method to the obtained subspaces and discover functions that are jointly smooth on all manifolds. We show analytically that our method is guaranteed to provide a set of orthogonal functions that are as jointly smooth as possible, ordered by increasing Dirichlet energy from the smoothest to the least smooth. In addition, we show that the extracted functions can be efficiently extended to unseen data using the Nyström method. We demonstrate the proposed method on both simulated and real measured data and compare the results to nonlinear variants of the seminal Canonical Correlation Analysis (CCA). Particularly, we show superior results for sleep stage identification. In addition, we show how the proposed method can be leveraged for finding minimal realizations of parameter spaces of nonlinear dynamical systems.

Or Yair, Felix Dietrich, Ronen Talmon et al.

In this paper, we address the problem of Domain Adaptation (DA) using Optimal Transport (OT) on Riemannian manifolds. We model the difference between two domains by a diffeomorphism and use the polar factorization theorem to claim that OT is indeed optimal for DA in a well-defined sense, up to a volume preserving map. We then focus on the manifold of Symmetric and Positive-Definite (SPD) matrices, whose structure provided a useful context in recent applications. We demonstrate the polar factorization theorem on this manifold. Due to the uniqueness of the weighted Riemannian mean, and by exploiting existing regularized OT algorithms, we formulate a simple algorithm that maps the source domain to the target domain. We test our algorithm on two Brain-Computer Interface (BCI) data sets and observe state of the art performance.

Seungjoon Lee, Felix Dietrich, George E. Karniadakis et al.

In statistical modeling with Gaussian Process regression, it has been shown that combining (few) high-fidelity data with (many) low-fidelity data can enhance prediction accuracy, compared to prediction based on the few high-fidelity data only. Such information fusion techniques for multifidelity data commonly approach the high-fidelity model $f_h(t)$ as a function of two variables $(t,y)$, and then using $f_l(t)$ as the $y$ data. More generally, the high-fidelity model can be written as a function of several variables $(t,y_1,y_2....)$; the low-fidelity model $f_l$ and, say, some of its derivatives, can then be substituted for these variables. In this paper, we will explore mathematical algorithms for multifidelity information fusion that use such an approach towards improving the representation of the high-fidelity function with only a few training data points. Given that $f_h$ may not be a simple function -- and sometimes not even a function -- of $f_l$, we demonstrate that using additional functions of $t$, such as derivatives or shifts of $f_l$, can drastically improve the approximation of $f_h$ through Gaussian Processes. We also point out a connection with "embedology" techniques from topology and dynamical systems.