Invertible Koopman neural operator for data-driven modeling of partial differential equations

2025-03-25

AI Summary

Existing Koopman-based methods suffer from reconstruction loss due to imperfect observable function design and inaccurate inverse mapping, and exhibit low efficiency and poor generalization when modeling partial differential equations (PDEs). To address these limitations, we propose the Invertible Koopman Neural Operator (IKNO). First, IKNO employs an invertible neural network to jointly parameterize both the observable function and its exact inverse, enabling lossless state reconstruction. Second, it pioneers the integration of the Koopman linear structure into frequency-domain evolution modeling, explicitly learning low-frequency-dominant modes to ensure resolution independence. Third, it incorporates interpolation and dimensionality-expansion preprocessing to handle non-Cartesian and unstructured geometric domains. Across diverse PDE benchmarks, IKNO consistently outperforms existing neural operators, achieving superior accuracy, strong cross-resolution robustness, and enhanced out-of-distribution generalization.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Koopman operator theory is a popular candidate for data-driven modeling because it provides a global linearization representation for nonlinear dynamical systems. However, existing Koopman operator-based methods suffer from shortcomings in constructing the well-behaved observable function and its inverse and are inefficient enough when dealing with partial differential equations (PDEs). To address these issues, this paper proposes the Invertible Koopman Neural Operator (IKNO), a novel data-driven modeling approach inspired by the Koopman operator theory and neural operator. IKNO leverages an Invertible Neural Network to parameterize observable function and its inverse simultaneously under the same learnable parameters, explicitly guaranteeing the reconstruction relation, thus eliminating the dependency on the reconstruction loss, which is an essential improvement over the original Koopman Neural Operator (KNO). The structured linear matrix inspired by the Koopman operator theory is parameterized to learn the evolution of observables' low-frequency modes in the frequency space rather than directly in the observable space, sustaining IKNO is resolution-invariant like other neural operators. Moreover, with preprocessing such as interpolation and dimension expansion, IKNO can be extended to operator learning tasks defined on non-Cartesian domains. We fully support the above claims based on rich numerical and real-world examples and demonstrate the effectiveness of IKNO and superiority over other neural operators.

Problem

• Constructing well-behaved observable functions for Koopman operators
• Improving efficiency in modeling partial differential equations
• Extending operator learning to non-Cartesian domains

Innovation

• Invertible Neural Network for observable functions
• Frequency space learning for low-frequency modes
• Resolution-invariant operator learning on non-Cartesian domains