Learning Hidden Physics and System Parameters with Deep Operator Networks

TL;DR

This work tackles hidden-physics discovery and parameter identification from sparse observations by introducing two complementary operator-learning frameworks built on DeepONet: Deep Hidden Physics Operator (DHPO) for cross-PDE-family discovery of unknown terms, and a physics-informed inverse mapping that uses a pretrained DeepONet to reconstruct fields and infer parameters. DHPO integrates a branch/trunk DeepONet with a hidden-physics MLP to learn the operator $\mathcal{N}$ in $\partial u/\partial t = \mathcal{N}(u,u_x,u_{xx},...) + f(x)$ under IC/BC and PDE residual constraints, enabling interpretable, mechanistic insights. The parameter-identification framework combines forward surrogate learning with an inverse map to estimate unknown parameters, offering deterministic estimates and calibrated uncertainty via a probabilistic variant using reparameterization and KL regularization. Demonstrated on Reaction-Diffusion, Burgers', 2D Heat, and Helmholtz problems, the methods achieve relative solution errors around $10^{-2}$ and parameter errors around $10^{-3}$ under sparse/noisy data, providing a data-efficient path to robust inverse modeling and PDE discovery with substantial computational speedups over traditional iterative approaches.

Abstract

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.

Figures (38)

• Figure 1: Architecture of the proposed Deep Hidden Physics Operator framework developed to discover the unknown physics leveraging sparsely labeled dataset. The DeepONet learns the solution operator for varying system conditions to predict the desired output. The solution as well as the gradients are considered as inputs to the MLP and it predicts the unknown physics. The entire framework is trained in a single training session with physics loss as defined in Equation \ref{['eq:eqn_loss']} and the data loss.
• Figure 1: Loss function over the training process of optimal model: (a) Reaction diffusion equation with $N_{train} = 500, N_d = 500.$ (b) Burgers' equation with $N_{train} = 1000, N_d = 500.$ (c) Heat equation with $N_{train} = 500, N_d = 550.$
• Figure 2: Schematic representation of the proposed architecture for unknown system parameters identification, $\nu$ for the Burger's equation. In the schematic, $u^s$ denotes the known velocity field at fixed sensor locations. In step 1, a data-driven DeepONet is trained with loss defined at just the sensor locations to reconstruct the field based on sparse measurements. In the second step, the trained DeepONet is given a warm start, and the learning of the solution is improved in conjunction with an MLP which predicts the unknown $\nu$ employing a physics loss. $\bm\theta^*, \bm\phi^*$ are the optimized parameters of the networks.
• Figure 2: Sample 1: hidden physics solution comparison, mean absolute error = 0.03177
• Figure 3: Reaction–diffusion equation: reference vs. DHPO predictions and HPM-integrated trajectories for two representative test samples drawn from the sine-function input space. Training data spans $t\le1$, while the extrapolation region ($t>1$) is indicated by white horizontal lines. For Sample 1, mean absolute errors are $u_{pred}(t\le1)=0.00432$, $u_{pred}(t>1)=0.04360$, $u_{hpm}(t\le1)=0.00226$, $u_{hpm}(t>1)=0.01917$. For Sample 2, mean absolute errors are $u_{pred}(t\le1)=0.01299$, $u_{pred}(t>1)=0.12473$, $u_{hpm}(t\le1)=0.00655$, $u_{hpm}(t>1)=0.05840$.