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

arXiv: https://arxiv.org/abs/2503.19717 · HTML: https://mlsanity.com/paper/2503.19717/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 101 references that are in the map.
- First posted on arXiv 25 Mar 2025.

## Builds on (most cited first, 30 of 46)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [On the training and generalization of deep operator networks](https://mlsanity.com/paper/2309.01020/) (2023)
- [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013)
- [Learning stiff chemical kinetics using extended deep neural operators](https://mlsanity.com/paper/2302.12645/) (2023)
- [Koopman neural operator as a mesh-free solver of non-linear partial differential equations](https://mlsanity.com/paper/2301.10022/) (2023)
- [Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks](https://mlsanity.com/paper/2008.05730/) (2020)
- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)
- [Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains](https://mlsanity.com/paper/2305.19663/) (2023)
- [Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2501.13329/) (2025)
- [Deep Learning Enhanced Dynamic Mode Decomposition](https://mlsanity.com/paper/2108.04433/) (2021)
- [A Review of Data-Driven Discovery for Dynamic Systems](https://mlsanity.com/paper/2210.10663/) (2022)

## Built on by (most cited first, 1 of 1)

- [Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators](https://mlsanity.com/paper/2605.10154/) (2026)

## Abstract (arXiv)

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.