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

This paper introduces Inelastic Constitutive Kolmogorov-Arnold Networks (iCKANs), a novel machine learning framework that automatically discovers interpretable, closed-form symbolic constitutive laws for both elastic and inelastic material behaviors by translating experimental data into potential functions, as demonstrated on viscoelastic polymers.

The Gist

Imagine you are trying to figure out the "personality" of a new material, like a super-stretchy rubber band or a soft gel. In engineering, we call this finding the constitutive law. It's basically the rulebook that says: "If I pull you this hard, you will stretch this much. If I hold you there, you will slowly relax. If I heat you up, you will get softer."

Traditionally, scientists have to guess these rules by hand, writing complex math equations based on their intuition. It's like trying to write a novel by guessing every word without reading the story first. It's slow, hard, and often misses the nuances of how real materials actually behave.

This paper introduces a new, smarter way to do this called iCKANs (Inelastic Constitutive Kolmogorov-Arnold Networks). Here is how it works, explained simply:

1. The Problem: The "Black Box" vs. The "Gibberish"

• Old Way (Traditional Math): Scientists write a specific equation. It's easy to read, but if the material is weird (like a rubber that gets squishy when hot), the equation might be wrong.
• New Way (Standard AI): You feed data into a giant neural network (a "Black Box"). It learns the material perfectly and predicts the future accurately. But, the result is a giant, unreadable mess of numbers. You can't ask the AI why it thinks the material behaves that way. It's like having a GPS that tells you exactly where to turn but refuses to show you the map.

2. The Solution: The "Smart Translator" (iCKAN)

The authors created a new type of AI that acts like a Smart Translator. It learns the material's behavior just as well as the "Black Box," but instead of keeping the secret, it translates its learning into clean, readable math equations that humans can understand.

Think of it like this:

• The Black Box is a genius chef who makes a perfect dish but won't tell you the recipe.
• The iCKAN is a genius chef who makes the perfect dish and writes down the recipe in plain English so you can cook it yourself.

3. How It Works: The "Spring and Sponge" Analogy

To understand the material, the iCKAN splits the problem into two parts, like a mechanical toy made of a Spring and a Sponge:

• The Spring (Elastic Part): When you pull it, it stretches immediately and snaps back. This is the "instant" reaction.
• The Sponge (Inelastic Part): When you pull it, it squishes slowly and takes time to recover. This is the "delayed" reaction (like memory foam).

The iCKAN learns two separate "rulebooks" (mathematical functions):

1. One for the Spring (how it snaps).
2. One for the Sponge (how it flows).

4. The Secret Sauce: "Symbolic Regression"

Here is the magic trick. Most AI models are made of layers of neurons that are hard to interpret. The iCKAN uses a special architecture (called a Kolmogorov-Arnold Network) where the "neurons" are actually flexible curves (B-Splines).

Once the AI has learned the material's behavior, the researchers use a process called Symbolic Regression. Imagine the AI has drawn a squiggly line on a graph. The Symbolic Regression looks at that squiggly line and asks: "What simple math formula looks exactly like this?"

It might say: "Oh, this squiggly line is actually just y = 2x + 5" or "This is y = x squared."

Suddenly, the complex AI behavior is turned into a simple, elegant equation that a human engineer can read, understand, and use in their own simulations.

5. Why This Matters: The "Weather Forecast" for Materials

The paper tested this on real rubber polymers (VHB 4910 and 4905).

• The Result: The iCKAN didn't just predict the stress correctly; it discovered the actual mathematical formulas that describe the rubber.
• The Superpower: It could also handle extra information, like temperature. If you tell the AI, "Hey, it's hot outside," it learns how the rubber changes. It then writes a new equation that includes temperature as a variable.

The Analogy:
Imagine you are teaching a robot to drive a car.

• Standard AI: The robot drives perfectly but crashes if the road gets icy because it doesn't understand why ice is slippery.
• iCKAN: The robot drives perfectly, and then it writes a manual: "When the road is wet, friction drops by 20%. When it's hot, the tires get softer." Now, any human can read that manual and understand the physics.

Summary

This paper presents a tool that bridges the gap between Data-Driven AI (which is powerful but opaque) and Physics-Based Modeling (which is clear but rigid).

iCKANs are like a detective that:

1. Looks at the clues (experimental data).
2. Solves the mystery (learns the material's behavior).
3. Writes a clear, readable report (a simple math equation) explaining exactly how the material works, including how it reacts to heat, stretching, and time.

This allows engineers to discover new material laws automatically, without needing to be math wizards, while still keeping the "why" and "how" of the physics intact.

Technical Summary

1. Problem Statement

The identification of constitutive laws (the relationship between stress and strain) for complex materials, particularly those exhibiting inelastic behavior (e.g., viscoelasticity, plasticity) at finite strains, remains a significant challenge in computational mechanics.

• Limitations of Traditional Models: Conventional models rely on simplifying assumptions that often fail to capture the full complexity of real-world material behavior. Developing them is labor-intensive, incremental, and requires deep domain expertise.
• Limitations of Existing Data-Driven Approaches:
  • Direct Data-Driven Methods: Often lack closed-form expressions, making them difficult to integrate into standard finite element solvers.
  • Symbolic Regression: Can struggle with high-dimensional data and complex inelastic histories.
  • Standard Neural Networks (e.g., CANNs, PANNs): While flexible, they often act as "black boxes," lacking physical interpretability. They may also struggle to enforce strict thermodynamic constraints (like convexity of potentials) without complex loss function penalties.
• The Gap: There is a need for a framework that combines the predictive accuracy of deep learning, the thermodynamic consistency of continuum mechanics, and the interpretability of symbolic mathematics, specifically for inelastic materials.

2. Methodology: Inelastic Constitutive Kolmogorov-Arnold Networks (iCKANs)

The authors propose iCKANs, a novel architecture that integrates the Generalized Standard Materials (GSM) framework with Kolmogorov-Arnold Networks (KANs).

A. Theoretical Foundation (Constitutive Framework)

The model is built upon the multiplicative decomposition of the deformation gradient ($\mathbf{F} = \mathbf{F}_e \mathbf{F}_i$) into elastic and inelastic parts. The material behavior is governed by two scalar potentials:

1. Elastic Potential ($\psi$): Depends on the elastic right Cauchy-Green tensor ($\bar{\mathbf{C}}_e$).
2. Inelastic Potential ($\omega$): A dissipation potential depending on the Mandel-like stress ($\bar{\mathbf{\Sigma}}$).
   The framework ensures thermodynamic consistency by enforcing that $\psi$ and $\omega$ are convex and satisfy specific boundary conditions (e.g., zero energy/stress at the undeformed state).

B. Network Architecture (KANs)

Unlike Multi-Layer Perceptrons (MLPs) that use fixed activation functions on linear layers, KANs replace linear weights with trainable univariate activation functions (B-Splines) on the edges.

• Input-Convexity: To satisfy thermodynamic laws, the authors design Partially Input-Convex KANs.
  • The elastic potential $\psi$ is modeled as convex with respect to strain invariants.
  • The inelastic potential $\omega$ is modeled as convex with respect to stress invariants.
  • Non-mechanical features (e.g., temperature) are included as inputs but are not constrained to be convex.
• Post-Processing: To ensure the inelastic potential is zero-valued at the origin and non-negative, a specific H-operator is applied to the network output: $\tilde{f}(x) = f(x) - f(\alpha) - \nabla f(\alpha) \cdot (x-\alpha)$.

C. Symbolic Extraction (The "Discovery" Phase)

A key feature of iCKANs is the ability to convert the trained neural network back into human-readable mathematics:

1. Sparsification: $L_1$ regularization and pruning remove unnecessary connections.
2. Symbolification: The trained B-Spline activations are approximated by a library of analytical functions (e.g., polynomials, exponentials, logarithms) using symbolic regression.
3. Result: The final output is a closed-form symbolic equation for $\psi$ and $\omega$, rather than a black-box network.

D. Numerical Implementation

• Time Integration: The evolution of internal variables is solved using explicit time integration (preferred for efficiency) or implicit integration (using a Liquid Time-Constant Network helper for stability).
• Feature Augmentation: The framework allows arbitrary non-mechanical features (like temperature) to be fed into the potentials, enabling the discovery of thermo-viscoelastic laws.

3. Key Contributions

1. Generalized Framework for Inelasticity: Extends previous KAN-based work (which focused on hyperelasticity) to inelastic materials (viscoelasticity, plasticity) by incorporating internal variables and dissipation potentials.
2. Automated Model Discovery: The system automatically discovers the mathematical form of both elastic and inelastic potentials directly from data, without pre-defining the functional form.
3. Interpretability via Symbolic Regression: Unlike standard neural networks, iCKANs yield interpretable, closed-form symbolic expressions that preserve physical constraints (convexity, objectivity).
4. Handling Non-Mechanical Features: The framework successfully integrates arbitrary features (e.g., temperature) into the constitutive law, allowing the discovery of how environmental conditions affect material properties.
5. Thermodynamic Consistency: By enforcing convexity through the network architecture (rather than just loss penalties), the model guarantees physical validity across the entire input domain.

4. Results

The authors validated iCKANs on both synthetic and experimental datasets:

• Synthetic Data (Verification):

  • Trained on data generated from known analytical laws (Neo-Hookean elastic + quadratic inelastic).
  • Outcome: The iCKAN successfully recovered the exact symbolic forms of the potentials with high accuracy, demonstrating the method's ability to "rediscover" known physics. Both explicit and implicit time integration schemes yielded comparable results.

• Experimental Data (VHB 4910 Polymer):

  • Task: Discover a viscoelastic model for a highly nonlinear polymer under cyclic loading.
  • Outcome: The model used two parallel branches (Maxwell-like elements) to capture complex hysteresis. The discovered symbolic expressions accurately predicted stress-strain curves for unseen strain rates and stretch levels.

• Experimental Data (VHB 4905 Thermo-Viscoelastic Polymer):

  • Task: Discover a model dependent on temperature ($0^\circ\text{C}$ to $80^\circ\text{C}$).
  • Outcome: The iCKAN successfully learned the temperature dependence.
    • The elastic potential was discovered as a function of strain invariants and a symbolic temperature function $g(\theta)$.
    • The inelastic potential was also successfully modeled.
  • Significance: The model provided explicit mathematical expressions showing exactly how temperature modifies the material's stiffness and dissipation, a capability difficult to achieve with standard neural networks.

5. Significance and Impact

• Bridging the Gap: iCKANs resolve the trade-off between the flexibility/accuracy of deep learning and the interpretability/physical consistency of classical mechanics.
• Efficiency: The final symbolic models are lightweight and can be directly implemented in finite element solvers without the computational overhead of evaluating a neural network during simulation.
• Future Applications: This framework paves the way for automated discovery of complex material laws for:
  • Plasticity and damage mechanics.
  • Biological tissues (growth and remodeling).
  • Anisotropic and composite materials.
  • Design of metamaterials with specific inelastic responses.
• Scientific Insight: By outputting symbolic equations, iCKANs do not just predict data; they provide physical insights (e.g., identifying specific functional dependencies on temperature or strain rate) that can guide material scientists in understanding underlying mechanisms.

In conclusion, the paper presents a robust, physics-informed machine learning framework that automates the discovery of interpretable constitutive laws for complex inelastic materials, marking a significant step forward in data-driven computational mechanics.