Sebastian J. Wetzel

Zakaria Patel, Ejaaz Merali, Sebastian J. Wetzel

We introduce an unsupervised machine learning method based on Siamese Neural Networks (SNN) to detect phase boundaries. This method is applied to Monte-Carlo simulations of Ising-type systems and Rydberg atom arrays. In both cases the SNN reveals phase boundaries consistent with prior research. The combination of leveraging the power of feed-forward neural networks, unsupervised learning and the ability to learn about multiple phases without knowing about their existence provides a powerful method to explore new and unknown phases of matter.

Sebastian J. Wetzel

Twinned regression methods are designed to solve the dual problem to the original regression problem, predicting differences between regression targets rather then the targets themselves. A solution to the original regression problem can be obtained by ensembling predicted differences between the targets of an unknown data point and multiple known anchor data points. We explore different aspects of twinned regression methods: (1) We decompose different steps in twinned regression algorithms and examine their contributions to the final performance, (2) We examine the intrinsic ensemble quality, (3) We combine twin neural network regression with k-nearest neighbor regression to design a more accurate and efficient regression method, and (4) we develop a simplified semi-supervised regression scheme.

Sebastian J. Wetzel, Zakaria Patel

It has been demonstrated that artificial neural networks like autoencoders or Siamese networks encode meaningful concepts in their latent spaces. However, there does not exist a comprehensive framework for retrieving this information in a human-readable form without prior knowledge. In quantitative disciplines concepts are typically formulated as equations. Hence, in order to extract these concepts, we introduce a framework for finding closed-form interpretations of neurons in latent spaces of artificial neural networks. The interpretation framework is based on embedding trained neural networks into an equivalence class of functions that encode the same concept. We interpret these neural networks by finding an intersection between the equivalence class and human-readable equations defined by a symbolic search space. Computationally, this framework is based on finding a symbolic expression whose normalized gradients match the normalized gradients of a specific neuron with respect to the input variables. The effectiveness of our approach is demonstrated by retrieving invariants of matrices and conserved quantities of dynamical systems from latent spaces of Siamese neural networks.

Sebastian J. Wetzel

Twin neural network regression is trained to predict differences between regression targets rather than the targets themselves. A solution to the original regression problem can be obtained by ensembling predicted differences between the targets of an unknown data point and multiple known anchor data points. Choosing the anchors to be the nearest neighbors of the unknown data point leads to a neural network-based improvement of k-nearest neighbor regression. This algorithm is shown to outperform both neural networks and k-nearest neighbor regression on small to medium-sized data sets.

Sebastian Johann Wetzel, Seungwoong Ha, Raban Iten et al.

Machine learning is increasingly transforming various scientific fields, enabled by advancements in computational power and access to large data sets from experiments and simulations. As artificial intelligence (AI) continues to grow in capability, these algorithms will enable many scientific discoveries beyond human capabilities. Since the primary goal of science is to understand the world around us, fully leveraging machine learning in scientific discovery requires models that are interpretable -- allowing experts to comprehend the concepts underlying machine-learned predictions. Successful interpretations increase trust in black-box methods, help reduce errors, allow for the improvement of the underlying models, enhance human-AI collaboration, and ultimately enable fully automated scientific discoveries that remain understandable to human scientists. This review examines the role of interpretability in machine learning applied to physics. We categorize different aspects of interpretability, discuss machine learning models in terms of both interpretability and performance, and explore the philosophical implications of interpretability in scientific inquiry. Additionally, we highlight recent advances in interpretable machine learning across many subfields of physics. By bridging boundaries between disciplines -- each with its own unique insights and challenges -- we aim to establish interpretable machine learning as a core research focus in science.

Sebastian Johann Wetzel

I introduce a unified framework for finding a closed-form interpretation of any single neuron in an artificial neural network. Using this framework I demonstrate how to interpret neural network classifiers to reveal closed-form expressions of the concepts encoded in their decision boundaries. In contrast to neural network-based regression, for classification, it is in general impossible to express the neural network in the form of a symbolic equation even if the neural network itself bases its classification on a quantity that can be written as a closed-form equation. The interpretation framework is based on embedding trained neural networks into an equivalence class of functions that encode the same concept. I interpret these neural networks by finding an intersection between the equivalence class and human-readable equations defined by a symbolic search space. The approach is not limited to classifiers or full neural networks and can be applied to arbitrary neurons in hidden layers or latent spaces.

Sebastian J. Wetzel, Roger G. Melko, Isaac Tamblyn

Twin neural network regression (TNNR) is a semi-supervised regression algorithm, it can be trained on unlabelled data points as long as other, labelled anchor data points, are present. TNNR is trained to predict differences between the target values of two different data points rather than the targets themselves. By ensembling predicted differences between the targets of an unseen data point and all training data points, it is possible to obtain a very accurate prediction for the original regression problem. Since any loop of predicted differences should sum to zero, loops can be supplied to the training data, even if the data points themselves within loops are unlabelled. Semi-supervised training improves TNNR performance, which is already state of the art, significantly.

Sebastian J. Wetzel, Kevin Ryczko, Roger G. Melko et al.

We introduce twin neural network (TNN) regression. This method predicts differences between the target values of two different data points rather than the targets themselves. The solution of a traditional regression problem is then obtained by averaging over an ensemble of all predicted differences between the targets of an unseen data point and all training data points. Whereas ensembles are normally costly to produce, TNN regression intrinsically creates an ensemble of predictions of twice the size of the training set while only training a single neural network. Since ensembles have been shown to be more accurate than single models this property naturally transfers to TNN regression. We show that TNNs are able to compete or yield more accurate predictions for different data sets, compared to other state-of-the-art methods. Furthermore, TNN regression is constrained by self-consistency conditions. We find that the violation of these conditions provides an estimate for the prediction uncertainty.

Dhananjay Ashok, Joseph Scott, Sebastian Wetzel et al.

We present a novel Auxiliary Truth enhanced Genetic Algorithm (GA) that uses logical or mathematical constraints as a means of data augmentation as well as to compute loss (in conjunction with the traditional MSE), with the aim of increasing both data efficiency and accuracy of symbolic regression (SR) algorithms. Our method, logic-guided genetic algorithm (LGGA), takes as input a set of labelled data points and auxiliary truths (ATs) (mathematical facts known a priori about the unknown function the regressor aims to learn) and outputs a specially generated and curated dataset that can be used with any SR method. Three key insights underpin our method: first, SR users often know simple ATs about the function they are trying to learn. Second, whenever an SR system produces a candidate equation inconsistent with these ATs, we can compute a counterexample to prove the inconsistency, and further, this counterexample may be used to augment the dataset and fed back to the SR system in a corrective feedback loop. Third, the value addition of these ATs is that their use in both the loss function and the data augmentation process leads to better rates of convergence, accuracy, and data efficiency. We evaluate LGGA against state-of-the-art SR tools, namely, Eureqa and TuringBot on 16 physics equations from "The Feynman Lectures on Physics" book. We find that using these SR tools in conjunction with LGGA results in them solving up to 30.0% more equations, needing only a fraction of the amount of data compared to the same tool without LGGA, i.e., resulting in up to a 61.9% improvement in data efficiency.

Lukas Kades, Jan M. Pawlowski, Alexander Rothkopf et al.

We explore artificial neural networks as a tool for the reconstruction of spectral functions from imaginary time Green's functions, a classic ill-conditioned inverse problem. Our ansatz is based on a supervised learning framework in which prior knowledge is encoded in the training data and the inverse transformation manifold is explicitly parametrised through a neural network. We systematically investigate this novel reconstruction approach, providing a detailed analysis of its performance on physically motivated mock data, and compare it to established methods of Bayesian inference. The reconstruction accuracy is found to be at least comparable, and potentially superior in particular at larger noise levels. We argue that the use of labelled training data in a supervised setting and the freedom in defining an optimisation objective are inherent advantages of the present approach and may lead to significant improvements over state-of-the-art methods in the future. Potential directions for further research are discussed in detail.

Sebastian Johann Wetzel

We employ unsupervised machine learning techniques to learn latent parameters which best describe states of the two-dimensional Ising model and the three-dimensional XY model. These methods range from principal component analysis to artificial neural network based variational autoencoders. The states are sampled using a Monte-Carlo simulation above and below the critical temperature. We find that the predicted latent parameters correspond to the known order parameters. The latent representation of the states of the models in question are clustered, which makes it possible to identify phases without prior knowledge of their existence or the underlying Hamiltonian. Furthermore, we find that the reconstruction loss function can be used as a universal identifier for phase transitions.