# Exactly conservative physics-informed neural networks and deep operator networks for dynamical systems

arXiv: https://arxiv.org/abs/2311.14131 · HTML: https://mlsanity.com/paper/2311.14131/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 42 references that are in the map.
- First posted on arXiv 23 Nov 2023.

## Builds on (most cited first, 17 of 17)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)
- [Structure preserving deep learning](https://mlsanity.com/paper/2006.03364/) (2020)
- [Improving physics-informed neural networks with meta-learned optimization](https://mlsanity.com/paper/2303.07127/) (2023)
- [Exact conservation laws for neural network integrators of dynamical systems](https://mlsanity.com/paper/2209.11661/) (2022)
- [Improving physics-informed DeepONets with hard constraints](https://mlsanity.com/paper/2309.07899/) (2023)

## Built on by (most cited first, 13 of 13)

- [Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space](https://mlsanity.com/paper/2501.12116/) (2025)
- [PinnDE: Physics-Informed Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2408.10011/) (2024)
- [Adaptive Correction for Ensuring Conservation Laws in Neural Operators](https://mlsanity.com/paper/2505.24579/) (2025)
- [Physically consistent model learning for reaction-diffusion systems](https://mlsanity.com/paper/2512.14240/) (2025)
- [FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates](https://mlsanity.com/paper/2602.01941/) (2026)
- [Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations](https://mlsanity.com/paper/2602.12706/) (2026)
- [Improving regional weather forecasts with neural interpolation](https://mlsanity.com/paper/2505.12040/) (2025)
- [Low-rank adaptive physics-informed HyperDeepONets for solving differential equations](https://mlsanity.com/paper/2507.18346/) (2025)
- [Physics-Informed Operator Learning for Hemodynamic Modeling](https://mlsanity.com/paper/2509.17293/) (2025)
- [Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks](https://mlsanity.com/paper/2511.09048/) (2025)
- [A Comparative Investigation of Thermodynamic Structure-Informed Neural Networks](https://mlsanity.com/paper/2603.26803/) (2026)
- [Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNs](https://mlsanity.com/paper/2603.29237/) (2026)
- [A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws](https://mlsanity.com/paper/2604.15617/) (2026)

## Abstract (arXiv)

We introduce a method for training exactly conservative physics-informed neural networks and physics-informed deep operator networks for dynamical systems. The method employs a projection-based technique that maps a candidate solution learned by the neural network solver for any given dynamical system possessing at least one first integral onto an invariant manifold. We illustrate that exactly conservative physics-informed neural network solvers and physics-informed deep operator networks for dynamical systems vastly outperform their non-conservative counterparts for several real-world problems from the mathematical sciences.