Neural Constitutive Learning for Generalized Reaction-Diffusion Systems

2026-09-29

AI Summary

This study addresses the challenge of unifying diverse reaction-diffusion mechanisms within neural PDE solvers by proposing the NCL-MCT solver. The method introduces a trajectory-agnostic constitutive learning paradigm that integrates a Mass Compression Transport (MCT) integrator with neural constitutive laws, enabling the extraction of system-specific constitutive responses through a shared integrator. Training is supervised by known physical laws and efficiently optimizes the model without requiring full solution trajectories. Experimental results demonstrate that the proposed solver achieves relative errors ranging from 10⁻⁴ to 10⁻² across seven classes of reaction-diffusion systems. These findings validate its generalization and reusability across varying initial conditions and temporal domains, establishing a new paradigm for unified modeling of multiple dynamical systems.

Abstract

Generalized reaction-diffusion systems encompass diverse transport mechanisms and coupled reaction kinetics. A central question for neural PDE solvers is what should be learned so that a common interface can accommodate phase-field and degenerate transport, local reactions, and multispecies coupling. We propose the Neural Constitutive Laws--Mass-Compression-Transport (NCL-MCT) Solver, which learns PDE-specific constitutive responses while retaining temporal evolution in a shared MCT integrator. Transport is represented through mobility and thermodynamic driving force, and reaction through relative reaction rates. These constitutive responses depend on the current density rather than explicitly on the initial condition or elapsed time, motivating their reuse across different initial conditions and time horizons. The same interface supports velocity-data supervision and known-law supervision, neither of which requires time integration during training. When constitutive laws are known, supervision can be evaluated on independently sampled density fields, enabling trajectory-free constitutive learning without generating solution trajectories. Across seven systems, separately trained constitutive modules share the same interface and MCT integrator and achieve relative rollout $L^2$ errors of $10^{-4}$ to $10^{-2}$. Tests with unseen initial-condition families and an extended time horizon assess reuse beyond training conditions, while separate experiments demonstrate trajectory-free constitutive learning. These results support constitutive responses as an effective learning target for a shared neural PDE framework.

Problem

• reaction-diffusion systems
• neural PDE solvers
• constitutive learning
• generalized transport

Innovation

• Neural Constitutive Laws
• Reaction-Diffusion Systems
• Trajectory-free Learning
• Mass-Compression-Transport Integrator
• Neural PDE Solver