Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models

TL;DR

This novel artificial neural network architecture can discover in an automated manner symbolic constitutive laws describing both the elastic and inelastic behavior of materials, and makes iCKANs a powerful tool to discover in the future how specific processing or service conditions affect the properties of materials.

Abstract

A key problem of solid mechanics is the identification of the constitutive law of a material, that is, the relation between strain and stress. Machine learning has lead to considerable advances in this field lately. Here we introduce inelastic Constitutive Kolmogorov-Arnold Networks (iCKANs). This novel artificial neural network architecture can discover in an automated manner symbolic constitutive laws describing both the elastic and inelastic behavior of materials. That is, it can translate data from material testing into corresponding elastic and inelastic potential functions in closed mathematical form. We demonstrate the advantages of iCKANs using both synthetic data and experimental data of the viscoelastic polymer materials VHB 4910 and VHB 4905. The results demonstrate that iCKANs accurately capture complex viscoelastic behavior while preserving physical interpretability. It is a particular strength of iCKANs that they can process not only mechanical data but also arbitrary additional information available about a material (e.g., about temperature-dependent behavior). This makes iCKANs a powerful tool to discover in the future also how specific processing or service conditions affect the properties of materials.

Figures (18)

• Figure 1: Inelastic Constitutive Kolmogorov-Arnold Network (iCKAN) pipeline for automated interpretable model discovery of inelastic materials. Three-dimensional stress-strain data with optional additional features (e.g., temperature) collected in a feature vector $\mathbf{f}$ are used to train the iCKAN model, which consists of two KANs representing the elastic potential $\psi$ and the inelastic potential $\omega$. The trained model can then be analyzed using symbolic regression to extract interpretable mathematical expressions.
• Figure 2: Multiplicative split of the deformation gradient into elastic and inelastic part, including the non-uniqueness of the intermediate configuration and the co-rotated intermediate configuration holthusen2023inelastic.
• Figure 3: (a) Demonstration of the postprocessing of a convex and non-decreasing function $f(x)$ to achieve a general convex function with the designed stationary point at $x=0$. $\tilde{f}(x)$ is achieved by $\mathcal{H}$-operation in \ref{['eq:omega_conv']} and $\hat{f}(x)$ is achieved by \ref{['eq:omega_c']}. (b) A partially input-convex Kolmogorov-Arnold Network, where the output $z$ is convex with respect to the yellow-marked input variables $(x_1, x_2, x_3)$ and unconstrained to the blue-marked input variables $y_1$.
• Figure 4: Representation of the iCKAN formulation in the form of a Maxwell model. The elastic potential $\psi$ is associated with the spring, and the inelastic potential $\omega$ with the dashpot.
• Figure 5: Explicit iCKAN architecture at timestep $t$. The inputs at each step are the current deformation gradient and time increment $(\mathbf{F}_t,\Delta t)$ together with the state variables $(\mathbf{C}_{t-1},\mathbf{U}_{i,t-1})$ from the previous step. The KAN models for the elastic potential $\psi$ and the inelastic potential $\omega$ are evaluated to update the state variables according to \ref{['eq:explicit']}. The updated state is then propagated to the next time step. In addition, the output stress $\mathbf{P}$ is computed by evaluating the elastic potential $\psi$ with the updated state variables.