Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions

TL;DR

Confronting the interpretability gap in elastoplastic constitutive modeling, the paper introduces a two-step framework that yields human-understandable yield relations from data. It first builds an expressive yet structured surrogate via a Quadratic Neural Model (QNM) that learns a polynomial feature-space with univariate shape functions $f_i(x_i)$, enabling second-order interactions, and then applies symbolic regression to recover analytical expressions for those shape functions. The divide-and-conquer SR—one-dimensional searches per shape function—reduces the combinatorial complexity of multivariate SR and yields compact, interpretable equations for the yield surface, such as $f_y(\boldsymbol{\sigma})=0$, suitable for UMAT–FEM integration. Across synthetic, pressure-sensitive, and porous-metal datasets, the approach achieves accurate yield surfaces, reveals convexity and symmetry properties, and supports portable, analytically expressed constitutive laws with open-source code for verification.

Abstract

Conventional neural network elastoplasticity models are often perceived as lacking interpretability. This paper introduces a two-step machine learning approach that returns mathematical models interpretable by human experts. In particular, we introduce a surrogate model where yield surfaces are expressed in terms of a set of single-variable feature mappings obtained from supervised learning. A post-processing step is then used to re-interpret the set of single-variable neural network mapping functions into mathematical form through symbolic regression. This divide-and-conquer approach provides several important advantages. First, it enables us to overcome the scaling issue of symbolic regression algorithms. From a practical perspective, it enhances the portability of learned models for partial differential equation solvers written in different programming languages. Finally, it enables us to have a concrete understanding of the attributes of the materials, such as convexity and symmetries of models, through automated derivations and reasoning. Numerical examples have been provided, along with an open-source code to enable third-party validation.

Figures (7)

• Figure 1: The trade-off between expressivity and interpretability in various machine learning models. We introduced the Quadratic Neural Model (QNM), which enhances the expressivity of the Neural Additive Model (NAM) at the expense of reducing interpretability. However, by obtaining an analytical expression of the feature space mapping, we can achieve a better trade-off between expressivity and interpretability.
• Figure 2: The neural quadratic method for enhanced expressibility. Instead of using a fully connected neural network with a multi-dimensional input layer, the proposed univariate neural networks are trained to create feature space, forming the basis to express the yield function analytically. To enhance expressivity, additional bases founded by the product of features are introduced.
• Figure 3: Neural network architecture for each shape function. A Fourier layer is utilized to improve the training of classical MLP.
• Figure 4: Network expressivity: (a) vanilla single-layer MLP with 80 hidden neurons and (b) single-layer spectral layer with 40 hidden neurons. Both models are trained for 10,000 epochs with ADAM optimizer. In this demonstration example, the data is intentionally over-fitted to test the expressivity power of the spectral layer for complicated data.
• Figure 5: The divide-and-conquer symbolic regression for enhanced interpretability. A series of 1D symbolic regressions are trained to replace the 1D neural network basis function to form an analytical yield function.