Compositional Neural Operators for Multi-Dimensional Fluid Dynamics

TL;DR

This paper introduces Compositional Neural Operators, a modular framework that decomposes complex PDEs into pretrained physics-based blocks, improving adaptability, interpretability, and reusability in fluid dynamics modeling.

Contribution

The paper proposes a novel compositional approach using pretrained neural operator blocks for PDEs, enabling better generalization and interpretability in fluid dynamics applications.

Findings

The framework effectively models convection-diffusion, Burgers', and Navier-Stokes equations.

Pretrained elementary operators enhance adaptability to new physical systems.

Modular design improves interpretability and reusability of neural operators.

Abstract

Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim to provide universal surrogates, typical encoding-decoding approaches suffer from high pretraining costs and limited interpretability. In this paper, we propose Compositional Neural Operators (CompNO) for 2D systems, a framework that decomposes complex PDEs into a library of Foundation Blocks. Each block is a specialized Neural Operator pretrained on elementary physics. This modular library contains convection, diffusion, and nonlinear convection blocks as well as a Poisson Solver, enabling the framework to address the pressure-velocity coupling. These experts are assembled via an Adaptation Block featuring an Aggregator. This aggregator learns nonlinear interactions by minimizing data loss and physics-based residuals driven from governing equations. The proposed approach has been evaluated on the Convection-Diffusion equation, the Burgers' equation, and the Incompressible Navier-Stokes equation. Our results demonstrate that learning from elementary operators significantly improves adaptability, enhances model interpretability and facilitates the reuse of pretrained blocks when adapting to new physical systems.