An Exterior-Embedding Neural Operator Framework for Preserving Conservation Laws

TL;DR

Neural operators for PDEs often violate conservation laws, causing drift in $E(t)=\int_{\Omega} u(\mathbf{x},t)\,d\mathbf{x}$ and harming cross-domain generalization. The authors introduce the Exterior-Embedded Conservation Framework (ECF), a universal plug-in that enforces conservation by correcting spectral predictions in Fourier space via a Conserved Quantity Encoder and Conserved Quantity Decoder, with integrated ($ECF_{\mathcal{I}}$) or staged ($ECF_{\mathcal{S}}$) training modes. They provide a theoretical connection between conservation errors and RMSE, showing strict error reduction without sacrificing expressiveness, and validate the approach across six conservation-law PDE benchmarks (e.g., adiabatic, shallow-water, Allen-Cahn), achieving substantial gains. The framework offers a practical, architecture-agnostic path to physically consistent neural operators with modest overhead, enabling reliable long-horizon PDE simulations.

Abstract

Neural operators have demonstrated considerable effectiveness in accelerating the solution of time-dependent partial differential equations (PDEs) by directly learning governing physical laws from data. However, for PDEs governed by conservation laws(e.g., conservation of mass, energy, or matter), existing neural operators fail to satisfy conservation properties, which leads to degraded model performance and limited generalizability. Moreover, we observe that distinct PDE problems generally require different optimal neural network architectures. This finding underscores the inherent limitations of specialized models in generalizing across diverse problem domains. To address these limitations, we propose Exterior-Embedded Conservation Framework (ECF), a universal conserving framework that can be integrated with various data-driven neural operators to enforce conservation laws strictly in predictions. The framework consists of two key components: a conservation quantity encoder that extracts conserved quantities from input data, and a conservation quantity decoder that adjusts the neural operator's predictions using these quantities to ensure strict conservation compliance in the final output. Since our architecture enforces conservation laws, we theoretically prove that it enhances model performance. To validate the performance of our method, we conduct experiments on multiple conservation-law-constrained PDE scenarios, including adiabatic systems, shallow water equations, and the Allen-Cahn problem. These baselines demonstrate that our method effectively improves model accuracy while strictly enforcing conservation laws in the predictions.

Figures (4)

• Figure 1: (a) demonstrates the visualization of the CNO model's final prediction results on the Allen-Cahn equation (AC-DW) dataset. The first row presents the global prediction results, while the second row displays the corresponding central local region predictions, with deeper blue shades indicating smaller numerical values. (b) presents the predicted conserved quantities of both the CNO model and its corresponding embedded framework over 20 time steps, along with the relative error variations of these quantities compared to their initial states.
• Figure 2: The predictive results of five models on four different datasets were evaluated using relative mean squared error (RMSE), where smaller errors indicate better model performance.
• Figure 3: Overview of the ECF architecture. The Conserved Quantity Encoder$\mathcal{P}$ extracts the zero-frequency signal $\hat{\bm{c}}_{\bm{0}}$ from input data via Fourier transform $\mathcal{F}$. The Conserved Quantity Decoder$\mathcal{Q}$ replaces $\bm{c}_{\bm{0}}$ in the Neural Operator's predictions with $\hat{\bm{c}}_{\bm{0}}$ and obtains the final output through inverse Fourier transform $\mathcal{F}^{-1}$. The Neural Operator in the diagram can represent various types of neural operators.
• Figure 4: Time-evolution of relative conservation errors for baseline Transolver and its corresponding +ECF frameworks (+ECF$_{\mathcal{I}}$/+ECF$_{\mathcal{S}}$) in all datasets

Theorems & Definitions (6)

• Theorem 1
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• Theorem 3
• Theorem 1
• Theorem 2
• Theorem 3