Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs

TL;DR

ENO addresses learning solution operators for PDEs when explicit PDEs are unknown by incorporating energy-based inductive bias. It employs two nets—a solution-operator net $\mathcal{S}_{\bm{\theta}}$ and an energy net $\mathcal{H}_{\bm{\phi}}$—and introduces a gradient-flow penalty that enforces $\dot{\bm{u}} \approx \mathcal{G}\frac{\delta \mathcal{H}}{\delta \bm{u}}$. The approach is validated on Hamiltonian and dissipative PDEs, showing improved trajectory, energy, and mass predictions over baselines and enabling mesh-free, super-resolution inference without explicit PDEs. This framework broadens the applicability of physics-informed operator learning by leveraging energy conservation/dissipation priors to achieve data-efficient, physically consistent simulations, even when PDE forms are unknown.

Abstract

The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning of solution operators of partial differential equations (PDEs). However, these works still struggle to learn dynamics that obeys the laws of physics. This paper proposes Energy-consistent Neural Operators (ENOs), a general framework for learning solution operators of PDEs that follows the energy conservation or dissipation law from observed solution trajectories. We introduce a novel penalty function inspired by the energy-based theory of physics for training, in which the energy functional is modeled by another DNN, allowing one to bias the outputs of the DNN-based solution operators to ensure energetic consistency without explicit PDEs. Experiments on multiple physical systems show that ENO outperforms existing DNN models in predicting solutions from data, especially in super-resolution settings.

Figures (6)

• Figure 1: Intuitive view of gradient flows \ref{['eq:H_pde']}. Dashed lines represent contours of energy functional $\mathcal{H}$ on function space $\mathcal{U}$; blue dashed line represents low energy. Functional derivative $\delta\mathcal{H}/\delta\bm{u}$ is orthogonal to the contour at $\bm{u}$. Systems follow a flow $\dot{\bm{u}}_{\rm{cons.}}$ conserving $\mathcal{H}$ if $\mathcal{G}$ is skew-symmetric and a flow $\dot{\bm{u}}_{\rm{diss.}}$ dissipating $\mathcal{H}$ if $\mathcal{G}$ is negative (semi) definite.
• Figure 2: Basic idea of our proposed approach. Input functions are initial, boundary conditions, etc; output functions are the corresponding solutions. Goal is to obtain solution operator from input-output function pairs. Our aim is to introduce inductive biases such that the solution operator's output satisfies physical laws.
• Figure 3: Schematic sketch of the architecture and training loss of ENO. Red arrows indicate automatic differentiation. ENO contains two networks: operator net and energy net parameterize solution operator and energy functional, respectively. By simultaneously minimizing data loss $L(\bm{\theta})$ and penalty $\Omega(\bm{\theta},\bm{\phi})$ inspired by energy-based theory, we can obtain a solution operator (i.e., operator net) to predict a solution that follows energy conservation or dissipation law without explicit PDEs.
• Figure 4: Visualization of the predicted solutions. Right column for each system is the difference between ground truth and its prediction, where the difference values for KdV equation were multiplied by 4.
• Figure 5: Results for KdV equation ($(N_{\rm x}, N_{\rm t})=(10,10)$). First column is a visualization of predicted solutions. Second column is a difference between ground truth and its prediction, where the difference values were multiplied by 4. Third column provides a comparison between the true energy (black line) and its estimate (red line).

Theorems & Definitions (1)

• Theorem 3.1