# Learning Hidden Physics and System Parameters with Deep Operator Networks

arXiv: https://arxiv.org/abs/2412.05133 · HTML: https://mlsanity.com/paper/2412.05133/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 6 Dec 2024.

## Builds on (most cited first, 11 of 11)

- [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 networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [Causality-Respecting Adaptive Refinement for PINNs: Enabling Precise Interface Evolution in Phase Field Modeling](https://mlsanity.com/paper/2410.20212/) (2024)

## Built on by (most cited first, 7 of 7)

- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference](https://mlsanity.com/paper/2508.05190/) (2025)
- [MLPs and KANs for data-driven learning in physical problems: A performance comparison](https://mlsanity.com/paper/2504.11397/) (2025)
- [Multimodal Neural Operators for Real-Time Biomechanical Modelling of Traumatic Brain Injury](https://mlsanity.com/paper/2510.03248/) (2025)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Local gradient neural operator](https://mlsanity.com/paper/2609.07752/) (2026)
- [Inferring physical fields in coupled systems with unknown parameters from incomplete observations using physics-constrained attentive neural operators](https://mlsanity.com/paper/2610.05723/) (2026)

## Abstract (arXiv)

Discovering hidden physical laws and identifying governing system parameters from sparse observations are central challenges in computational science and engineering. Existing data-driven methods, such as physics-informed neural networks (PINNs) and sparse regression, are limited by their need for extensive retraining, sensitivity to noise, or inability to generalize across families of partial differential equations (PDEs). In this work, we introduce two complementary frameworks based on deep operator networks (DeepONet) to address these limitations. The first, termed the Deep Hidden Physics Operator (DHPO), extends hidden-physics modeling into the operator-learning paradigm, enabling the discovery of unknown PDE terms across diverse equation families by identifying the mapping of unknown physical operators. The second is a parameter identification framework that combines pretrained DeepONet with physics-informed inverse modeling to infer system parameters directly from sparse sensor data. We demonstrate the effectiveness of these approaches on benchmark problems, including the Reaction-Diffusion system, Burgers' equation, the 2D Heat equation, and 2D Helmholtz equation. Across all cases, the proposed methods achieve high accuracy, with relative solution errors on the order of O(10^-2) and parameter estimation errors on the order of O(10^-3), even under limited and noisy observations. By uniting operator learning with physics-informed modeling, this work offers a unified and data-efficient framework for physics discovery and parameter identification, paving the way for robust inverse modeling in complex dynamical systems.