# Factorized Neural Implicit DMD for Parametric Dynamics

arXiv: https://arxiv.org/abs/2603.10995 · HTML: https://mlsanity.com/paper/2603.10995/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 34 references that are in the map.
- First posted on arXiv 11 Mar 2026.

## Builds on (most cited first, 18 of 18)

- [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)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations](https://mlsanity.com/paper/2505.24717/) (2025)
- [Bagging, optimized dynamic mode decomposition (BOP-DMD) for robust, stable forecasting with spatial and temporal uncertainty-quantification](https://mlsanity.com/paper/2107.10878/) (2021)
- [DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction](https://mlsanity.com/paper/2504.19496/) (2025)
- [Zebra: In-Context Generative Pretraining for Solving Parametric PDEs](https://mlsanity.com/paper/2410.03437/) (2024)
- [Koopman operator learning using invertible neural networks](https://mlsanity.com/paper/2306.17396/) (2023)
- [ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals](https://mlsanity.com/paper/2501.00701/) (2025)
- [ENMA: Tokenwise Autoregression for Generative Neural PDE Operators](https://mlsanity.com/paper/2506.06158/) (2025)
- [Koopman AutoEncoder via Singular Value Decomposition for Data-Driven Long-Term Prediction](https://mlsanity.com/paper/2408.11303/) (2024)
- [Fast Subspace Fluid Simulation with a Temporally-Aware Basis](https://mlsanity.com/paper/2502.05339/) (2025)

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

- [WorldParticle: Unified World Simulation of Lagrangian Particle Dynamics via Transformer](https://mlsanity.com/paper/2605.15305/) (2026)

## Abstract (arXiv)

A data-driven, model-free approach to modeling the temporal evolution of physical systems mitigates the need for explicit knowledge of the governing equations. Even when physical priors such as partial differential equations are available, such systems often reside in high-dimensional state spaces and exhibit nonlinear dynamics, making traditional numerical solvers computationally expensive and ill-suited for real-time analysis and control. Consider the problem of learning a parametric flow of a dynamical system: with an initial field and a set of physical parameters, we aim to predict the system's evolution over time in a way that supports long-horizon rollouts, generalization to unseen parameters, and spectral analysis. We propose a physics-coded neural field parameterization of the Koopman operator's spectral decomposition. Unlike a physics-constrained neural field, which fits a single solution surface, and neural operators, which directly approximate the solution operator at fixed time horizons, our model learns a factorized flow operator that decouples spatial modes and temporal evolution. This structure exposes underlying eigenvalues, modes, and stability of the underlying physical process to enable stable long-term rollouts, interpolation across parameter spaces, and spectral analysis. We demonstrate the efficacy of our method on a range of dynamics problems, showcasing its ability to accurately predict complex spatiotemporal phenomena while providing insights into the system's dynamic behavior.