Interpretable neural equation learning for high-dimensional materials data

Philip Jacobson; Prasad Iyer; Remi Dingreville; Saaketh Desai

cs.LGOct 27, 2026

Interpretable neural equation learning for high-dimensional materials data

Authors: Philip Jacobson, Prasad Iyer, Remi Dingreville, Saaketh Desai

Organizations: Sandia National Laboratories, Livermore, CA, USA. · Center for Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque , NM, USA.

Abstract

Equation learning seeks to discover compact, interpretable equations from data using various machine-learning techniques. However, few workflows discover and interpret learned equations in sparse, high-dimensional materials datasets. We present nn-EQL, an equation-learning framework that generates candidate equations through iterative training, pruning, and retraining of a neural network with customizable, user-specified neuron-level activation functions. Candidate equations are ranked using a minimum description length (MDL)-inspired score, balancing error and sparsity, and diagnosed to assess term-wise contributions, input-scaled sensitivities, and uncertainties. We demonstrate nn-EQL on standard AI-Feynman benchmarks as well as multiscale high-dimensional materials datasets such as finite-element simulations of heterogeneous materials, and high-throughput experimental thin-film deposition data. nn-EQL integrates with tabular data and high-dimensional data via standard encoder models, learning equations and providing insight into learned terms. Looking forward, integrating physically interpretable representations, multimodal and multi-fidelity data, and adaptive data acquisition could enable equation learning to formulate and test mechanistic hypotheses.

Figures & tables

Figure 1: nn-EQL workflow. A set of candidate equations are generated by iteratively pruning a large neural network with customized activation functions. A set of diagnostic tools are then used for assessing the quality of the prediction, including term wise contributions, sensitivities, and uncertainty. Those metrics are ultimately used to determine the equation that best fits the data.

Figure 2: nn-EQL performance on a subset of the AI-Feynman benchmark. (a) Discovery of equations for 31 problems. Each point is the selected nn-EQL equation for one problem, plotted as normalized tree-edit distance (NED) to the ground truth versus learned equation complexity, as measured by tree size. All selected equations are accurate ( $R^{2}_{\mathrm{test}}>0.99$ ), and 19 of 31 exactly recover the ground-truth structure (NED $=0$ ). (b) Candidate family generated by iterative pruning for equation I.11.19. (c) Term-level diagnostics for two selected equations: III.17.37 (top) and II.37.1 (bottom), where green terms are terms present in the ground truth equation.

Figure 3: nn-EQL performance on atomistic simulations of Cu $\langle 110\rangle$ symmetric tilt grain boundaries. (a) Grain-boundary energy $\gamma_{\mathrm{GB}}$ versus misorientation angle $\theta$ for 96 atomistic configurations, with curves from independent PySR runs and from nn-EQL overlaid on the data. (b) Term-level diagnostics for the nn-EQL expression. (c) Equations discovered using various search parameters for PySR, alongside the equation discovered with the nn-EQL (bold), along with $R^{2}$ on test set.

Figure 4: Equation learning from high-dimensional Mechanical MNIST material fields. (a) Example $28\times 28$ material property field and corresponding strain energy response over the loading trajectory. The enlarged marker denotes the final loading step used as the prediction target. (b) Test $R^{2}$ for various equation learning methods, both in pixel space and in latent space. Inset shows predicted versus true final strain energy for the compact nn-EQL model, evaluated on 1 000 held out test data points. (c) Signed input-scaled sensitivities for two representative material fields. Red and blue regions indicate pixels that locally increase or decrease the corresponding latent variable or predicted strain energy.

Figure 5: Learning equations from experimental data relating X-ray diffractograms to hardness for nanocrystalline binary alloys. (a) (Left) Sample X-ray diffractograms ordered by increasing hardness. (Right) Correlations between hardness and human-identified key features of diffractograms, including peak locations, peak widths, and intensity ratio. (b) (Left) Term-wise contributions and sensitivities, highlighting dominant terms governing hardness. (Right) Hardness as a function of $z_{0},z_{1}$ , as learned by nn-EQL (heatmap, see Eq. ( 20 )) with experimental data overlaid (data points). (c) (Left) Interpreting the linear terms in the learned equation with local spearman correlations between latent variables and human-identified features. Spearman correlations are calculated locally by holding one variable (e.g. $z_{0}$ ) nearly constant, and varying $z_{1}$ . Correlations are evaluated at 10 percentile levels, establishing the spread of correlations in various hardness regimes. (Right) Interpreting microstructural features where the $\rm 4^{th}$ order term dominates the linear term in the learned equation.

Symbolic Regression

Symbolic regression searches data for compact mathematical expressions that balance predictive accuracy with interpretability.

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Hyperparameter selection for equation learning with biologically-informed neural networks

William Lavery, Jodie A. Cochrane, John T. Nardini, Sara Hamis

Department of Information Technology, Uppsala University, Sweden · Department of Mathematics and Statistics, The College of New Jersey, USA

Hyperparameter Optimization Physics-Informed ML PDE Discovery

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Biologically-informed neural networks (BINNs) have emerged as a flexible subclass of physics-informed neural networks (PINNs) for learning terms in partial differential equations from data. BINNs are particularly suited for biological systems, where the governing equations are highly nonlinear and only partially known a priori, and where data observations are often sparse, noisy, and incomplete. However, applying BINNs effectively in practice depends critically on hyperparameter selection, which remains a central challenge in equation-learning frameworks. Hyperparameters are often chosen heuristically and only cursorily documented, which limits the reproducibility of results and the transferability of methods. We present a diagnostic workflow for hyperparameter selection that can be used when the ground-truth equations are not known. The workflow is guided by three main questions: (1) Are the benefits of greater network capacity worth the cost? (2) Do more training epochs keep reducing the validation loss? (3) Do the learned terms stop changing as network capacity and training increase? We apply our workflow to synthetic systems of varying complexity with known ground truth, spanning diffusion and growth right-hand side terms and data ranging from 1D+t to 2D+t. We demonstrate that the validation loss generally follows the true error in the learned terms and distil practical rules of thumb for selecting hyperparameters in the BINN architecture. By providing a structured workflow, practical guidelines, and suggested starting values for hyperparameter selection, this work lowers the barrier to BINN-based equation learning.

Abstract

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