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Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws

Ning Liu, Yiming Fan, Xianyi Zeng, Milan Klöwer, Lu Zhang, Yue Yu

TL;DR

This paper addresses the challenge of data-efficient learning of physics-based dynamics while respecting fundamental conservation laws. It introduces clawNO, a neural operator architecture that guarantees mass/volume conservation by constructing divergence-free outputs through a differential-forms-based potential $u= * d\mu$ with $\mu$ skew-symmetric, paired with a precomputed differentiation layer to realize the necessary derivatives. The approach is architecture-agnostic and compatible with Fourier and graph-based NOs, enabling automatic enforcement of conservation laws across diverse domains, including incompressible NS, shallow-water dam breaks, atmospheric dynamics, and Mooney–Rivlin deformation. Empirically, clawNOs outperform state-of-the-art NO baselines, especially in small-data regimes, while maintaining physical consistency (low divergence) and offering broad applicability to hidden-physics discovery without requiring full governing PDEs.

Abstract

Neural operators (NOs) have emerged as effective tools for modeling complex physical systems in scientific machine learning. In NOs, a central characteristic is to learn the governing physical laws directly from data. In contrast to other machine learning applications, partial knowledge is often known a priori about the physical system at hand whereby quantities such as mass, energy and momentum are exactly conserved. Currently, NOs have to learn these conservation laws from data and can only approximately satisfy them due to finite training data and random noise. In this work, we introduce conservation law-encoded neural operators (clawNOs), a suite of NOs that endow inference with automatic satisfaction of such conservation laws. ClawNOs are built with a divergence-free prediction of the solution field, with which the continuity equation is automatically guaranteed. As a consequence, clawNOs are compliant with the most fundamental and ubiquitous conservation laws essential for correct physical consistency. As demonstrations, we consider a wide variety of scientific applications ranging from constitutive modeling of material deformation, incompressible fluid dynamics, to atmospheric simulation. ClawNOs significantly outperform the state-of-the-art NOs in learning efficacy, especially in small-data regimes.

Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws

TL;DR

This paper addresses the challenge of data-efficient learning of physics-based dynamics while respecting fundamental conservation laws. It introduces clawNO, a neural operator architecture that guarantees mass/volume conservation by constructing divergence-free outputs through a differential-forms-based potential with skew-symmetric, paired with a precomputed differentiation layer to realize the necessary derivatives. The approach is architecture-agnostic and compatible with Fourier and graph-based NOs, enabling automatic enforcement of conservation laws across diverse domains, including incompressible NS, shallow-water dam breaks, atmospheric dynamics, and Mooney–Rivlin deformation. Empirically, clawNOs outperform state-of-the-art NO baselines, especially in small-data regimes, while maintaining physical consistency (low divergence) and offering broad applicability to hidden-physics discovery without requiring full governing PDEs.

Abstract

Neural operators (NOs) have emerged as effective tools for modeling complex physical systems in scientific machine learning. In NOs, a central characteristic is to learn the governing physical laws directly from data. In contrast to other machine learning applications, partial knowledge is often known a priori about the physical system at hand whereby quantities such as mass, energy and momentum are exactly conserved. Currently, NOs have to learn these conservation laws from data and can only approximately satisfy them due to finite training data and random noise. In this work, we introduce conservation law-encoded neural operators (clawNOs), a suite of NOs that endow inference with automatic satisfaction of such conservation laws. ClawNOs are built with a divergence-free prediction of the solution field, with which the continuity equation is automatically guaranteed. As a consequence, clawNOs are compliant with the most fundamental and ubiquitous conservation laws essential for correct physical consistency. As demonstrations, we consider a wide variety of scientific applications ranging from constitutive modeling of material deformation, incompressible fluid dynamics, to atmospheric simulation. ClawNOs significantly outperform the state-of-the-art NOs in learning efficacy, especially in small-data regimes.
Paper Structure (24 sections, 2 theorems, 34 equations, 18 figures, 5 tables)

This paper contains 24 sections, 2 theorems, 34 equations, 18 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. ClawNO: Conservation Law Encoded Neural Operator
  4. Built-In Divergence-Free Prediction
  5. ClawNO Architecture and Implementation
  6. Experiments
  7. Incompressible Navier-Stokes Equation
  8. Radial Dam Break
  9. Atmospheric Modeling
  10. Constitutive Modeling of Material Deformation
  11. Conclusion
  12. Related concepts of differential forms
  13. Equivariance of the differentiation layer
  14. Data generation and training strategies
  15. Example 1 -- incompressible Navier-Stokes equation
  16. ...and 9 more sections

Key Result

Theorem 3.1

$\mathcal{D}$ is a $p(p-1)/2\times M\times M\times p$ tensor with its parameters given by: $\mathcal{D}[\mu]=\star d\mu^{(N^{(1)},\cdots,N^{(p)})}=[(\mathcal{D}[\mu])_1,\cdots,(\mathcal{D}[\mu])_p]^T$, where then the following error estimate holds true: if $\mu$ has $m-1$ continuous derivatives for some $m\geq 2$ and a $m-$th derivative of bounded variation. In the special case when $\mu$ is smoo

Figures (18)

  • Figure 1: Predictability demo in atmospheric modeling. While FNO plausibly learns the wave propagation patterns, its relative L2 error is twice as big as that of clawNO (cf. Section \ref{['sec:sw3d']}), especially in the vicinity of mountains (Fig. \ref{['fig:swm']}). ClawNOs automatically satisfy the conservation law and improve physical consistency.
  • Figure 2: Proposed clawNO architecture. We start from the input function $\mathbf{g}(x)$. After lifting, the high-dimensional latent representation goes through a series of iterative equivariant layers, then gets projected to a function space in the form of an antisymmetric matrix. Lastly, we employ numerical differentiation (layer $\mathcal{D}$ with pre-calculated weights) to obtain the target divergence-free output.
  • Figure 3: The data distribution of velocity in L2 norm, in the incompressible Navier-Stokes dataset. Left: the $x$ component of velocity. Right: the $y$ component of the velocity. Cases (a)-(c) represent the histogram of sample distributions in the small, medium and large data regimes, respectively, with blue representing the test samples and orange for the training samples. The per-sample relative L2 error on the test set is also plotted in (a), comparing clawGFNO (in navy) with GFNL (in yellow). This result demonstrates the improved the accuracy of clawNO, comparing to its counterpart, in small data regime.
  • Figure 4: The per-time-step prediction error on the test dataset of the incompressible Navier-Stokes case 1: ntrain=10 (left) and ntrain=1000 (right).
  • Figure 5: Demonstration of zero-shot super resolution of clawFNO using the radial dam break dataset, where clawFNO is trained on $32\times32$ spatial resolutions and directly make predictions on $128\times128$ grids.
  • ...and 13 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Theorem 3.2
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4