# Physically consistent model learning for reaction-diffusion systems

arXiv: https://arxiv.org/abs/2512.14240 · HTML: https://mlsanity.com/paper/2512.14240/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 80 references that are in the map.
- First posted on arXiv 16 Dec 2025.

## Builds on (most cited first, 24 of 24)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [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)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Achieving Conservation of Energy in Neural Network Emulators for Climate Modeling](https://mlsanity.com/paper/1906.06622/) (2019)
- [INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservation](https://mlsanity.com/paper/2212.14365/) (2022)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Exact conservation laws for neural network integrators of dynamical systems](https://mlsanity.com/paper/2209.11661/) (2022)
- [Exactly conservative physics-informed neural networks and deep operator networks for dynamical systems](https://mlsanity.com/paper/2311.14131/) (2023)
- [Symbolic Recovery of Differential Equations: The Identifiability Problem](https://mlsanity.com/paper/2210.08342/) (2022)
- [Learning-informed parameter identification in nonlinear time-dependent PDEs](https://mlsanity.com/paper/2202.10915/) (2022)
- [On uniqueness in structured model learning](https://mlsanity.com/paper/2410.22009/) (2024)
- [Robust identifiability for symbolic recovery of differential equations](https://mlsanity.com/paper/2410.09938/) (2024)
- [On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs](https://mlsanity.com/paper/2306.05185/) (2023)
- [Sequential bi-level regularized inversion with application to hidden reaction law discovery](https://mlsanity.com/paper/2409.03834/) (2024)
- [Learning System Parameters from Turing Patterns](https://mlsanity.com/paper/2108.08542/) (2021)
- [Adaptive Correction for Ensuring Conservation Laws in Neural Operators](https://mlsanity.com/paper/2505.24579/) (2025)

## Built on by (most cited first, 1 of 1)

- [Symbolic recovery of PDEs from measurement data](https://mlsanity.com/paper/2602.15603/) (2026)

## Abstract (arXiv)

This paper addresses the problem of learning reaction-diffusion (RD) systems from data while ensuring physical consistency and well-posedness of the learned models. Building on a regularization-based framework for structured model learning, we focus on learning parameterized reaction terms and investigate how to incorporate key physical properties, such as mass conservation and quasipositivity, directly into the learning process. Our main contributions are twofold: First, we propose techniques to systematically modify a given class of parameterized reaction terms such that the resulting terms inherently satisfy mass conservation and quasipositivity, ensuring that the learned RD systems preserve non-negativity and adhere to physical principles. These modifications also guarantee well-posedness of the resulting PDEs under additional regularity and growth conditions. Second, we extend existing theoretical results on regularization-based model learning to RD systems using these physically consistent reaction terms. Specifically, we prove that solutions to the learning problem converge to a unique, regularization-minimizing solution of a limit system even when conservation laws and quasipositivity are enforced. In addition, we provide approximation results for quasipositive functions, essential for constructing physically consistent parameterizations. These results advance the development of interpretable and reliable data-driven models for RD systems that align with fundamental physical laws.