# Improving physics-informed DeepONets with hard constraints

arXiv: https://arxiv.org/abs/2309.07899 · HTML: https://mlsanity.com/paper/2309.07899/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 27 references that are in the map.
- First posted on arXiv 14 Sep 2023.

## Builds on (most cited first, 16 of 16)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)
- [Improving physics-informed neural networks with meta-learned optimization](https://mlsanity.com/paper/2303.07127/) (2023)
- [Deep-OSG: Deep Learning of Operators in Semigroup](https://mlsanity.com/paper/2302.03358/) (2023)
- [M-ENIAC: A machine learning recreation of the first successful numerical weather forecasts](https://mlsanity.com/paper/2304.09070/) (2023)

## Built on by (most cited first, 9 of 9)

- [Exactly conservative physics-informed neural networks and deep operator networks for dynamical systems](https://mlsanity.com/paper/2311.14131/) (2023)
- [Neural Parameter Regression for Explicit Representations of PDE Solution Operators](https://mlsanity.com/paper/2403.12764/) (2024)
- [PinnDE: Physics-Informed Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2408.10011/) (2024)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [Neural Spline Operators for Risk Quantification in Stochastic Systems](https://mlsanity.com/paper/2508.20288/) (2025)
- [Fredholm Integral Equations Neural Operator (FIE-NO) for Data-Driven Boundary Value Problems](https://mlsanity.com/paper/2408.12389/) (2024)
- [Low-rank adaptive physics-informed HyperDeepONets for solving differential equations](https://mlsanity.com/paper/2507.18346/) (2025)
- [Deep Operator Networks for Surrogate Modeling of Cyclic Adsorption Processes with Varying Initial Conditions](https://mlsanity.com/paper/2601.09491/) (2026)
- [The limits of exactness: On the failure of automatic differentiation in physics-informed machine learning](https://mlsanity.com/paper/2609.33078/) (2026)

## Abstract (arXiv)

Current physics-informed (standard or deep operator) neural networks still rely on accurately learning the initial and/or boundary conditions of the system of differential equations they are solving. In contrast, standard numerical methods involve such conditions in computations without needing to learn them. In this study, we propose to improve current physics-informed deep learning strategies such that initial and/or boundary conditions do not need to be learned and are represented exactly in the predicted solution. Moreover, this method guarantees that when a deep operator network is applied multiple times to time-step a solution of an initial value problem, the resulting function is at least continuous.