DimINO: Dimension-Informed Neural Operator Learning

TL;DR

DimINO introduces a dimension-informed framework for neural operators that improves generalization and efficiency in solving PDEs, supported by theoretical guarantees and empirical performance gains.

Contribution

It proposes DimINO, a novel neural operator framework with dimension-aware components, offering universal approximation and invariance properties, reducing model complexity.

Findings

Achieves up to 76.3% performance improvement on PDE datasets

Establishes a universal approximation theorem for DimINO

Demonstrates the STI property in empirical tests

Abstract

In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their ability to approximate operators-mappings between functions. Although neural operators benefit from a universal approximation theorem, achieving reliable error bounds often necessitates large model architectures, such as deep stacks of Fourier layers. This raises a natural question: Can we design lightweight models without sacrificing generalization? To address this, we introduce DimINO (Dimension-Informed Neural Operators), a framework inspired by dimensional analysis. DimINO incorporates two key components, DimNorm and a redimensionalization operation, which can be seamlessly integrated into existing neural operator architectures. These components enhance the model's ability to generalize across datasets with varying physical parameters. Theoretically, we establish a universal approximation theorem for DimINO and prove that it satisfies a critical property we term Similar Transformation Invariance (STI). Empirically, DimINO achieves up to 76.3% performance gain on PDE datasets while exhibiting clear evidence of the STI property.