# Neural Constitutive Learning for Generalized Reaction-Diffusion Systems

arXiv: https://arxiv.org/abs/2609.37113 · HTML: https://mlsanity.com/paper/2609.37113/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 47 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics](https://mlsanity.com/paper/2304.14369/) (2023)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Composing Partial Differential Equations with Physics-Aware Neural Networks](https://mlsanity.com/paper/2111.11798/) (2021)
- [Graph Neural PDE Solvers with Conservation and Similarity-Equivariance](https://mlsanity.com/paper/2405.16183/) (2024)
- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [UniPhy: Learning a Unified Constitutive Model for Inverse Physics Simulation](https://mlsanity.com/paper/2505.16971/) (2025)
- [Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems](https://mlsanity.com/paper/2412.03161/) (2024)
- [CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators](https://mlsanity.com/paper/2512.05297/) (2025)
- [Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference](https://mlsanity.com/paper/2508.05190/) (2025)
- [PAPM: A Physics-aware Proxy Model for Process Systems](https://mlsanity.com/paper/2407.05232/) (2024)
- [Unsupervised operator learning approach for dissipative equations via Onsager principle](https://mlsanity.com/paper/2508.07440/) (2025)
- [PC-NCLaws: Physics-Embedded Conditional Neural Constitutive Laws for Elastoplastic Materials](https://mlsanity.com/paper/2510.21404/) (2025)
- [Structure-Aware Variational Learning of a Class of Generalized Diffusions](https://mlsanity.com/paper/2604.20188/) (2026)

## Abstract (arXiv)

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 L2 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.