# Interpretable learning of effective dynamics for multiscale systems

arXiv: https://arxiv.org/abs/2309.05812 · HTML: https://mlsanity.com/paper/2309.05812/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 90 references that are in the map.
- First posted on arXiv 11 Sep 2023.

## Builds on (most cited first, 26 of 26)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Unsupervised Learning of Video Representations using LSTMs](https://mlsanity.com/paper/1502.04681/) (2015)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [A Clockwork RNN](https://mlsanity.com/paper/1402.3511/) (2014)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems](https://mlsanity.com/paper/1912.12976/) (2019)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [Physics-aware, probabilistic model order reduction with guaranteed stability](https://mlsanity.com/paper/2101.05834/) (2021)
- [CD-ROM: Complemented Deep-Reduced Order Model](https://mlsanity.com/paper/2202.10746/) (2022)
- [A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime](https://mlsanity.com/paper/1902.03968/) (2019)
- [Adaptive learning of effective dynamics: Adaptive real-time, online modeling for complex systems](https://mlsanity.com/paper/2304.01732/) (2023)
- [Interpretable reduced-order modeling with time-scale separation](https://mlsanity.com/paper/2303.02189/) (2023)

## Built on by (most cited first, 11 of 11)

- [Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations](https://mlsanity.com/paper/2408.00157/) (2024)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder](https://mlsanity.com/paper/2310.10745/) (2023)
- [Generative Learning for Forecasting the Dynamics of Complex Systems](https://mlsanity.com/paper/2402.17157/) (2024)
- [Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations](https://mlsanity.com/paper/2410.10174/) (2024)
- [Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learning](https://mlsanity.com/paper/2508.03926/) (2025)
- [A Mechanistic Analysis of Transformers for Dynamical Systems](https://mlsanity.com/paper/2512.21113/) (2025)
- [NeurIPS 2024 ML4CFD Competition: Results and Retrospective Analysis](https://mlsanity.com/paper/2506.08516/) (2025)
- [UP-dROM : Uncertainty-Aware and Parametrised dynamic Reduced-Order Model, application to unsteady flows](https://mlsanity.com/paper/2503.23236/) (2025)
- [Efficient Adaptation of ROMs for Unsteady Flows Using Data Assimilation](https://mlsanity.com/paper/2602.23188/) (2026)
- [Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometry](https://mlsanity.com/paper/2607.00669/) (2026)

## Abstract (arXiv)

The modeling and simulation of high-dimensional multiscale systems is a critical challenge across all areas of science and engineering. It is broadly believed that even with today's computer advances resolving all spatiotemporal scales described by the governing equations remains a remote target. This realization has prompted intense efforts to develop model order reduction techniques. In recent years, techniques based on deep recurrent neural networks have produced promising results for the modeling and simulation of complex spatiotemporal systems and offer large flexibility in model development as they can incorporate experimental and computational data. However, neural networks lack interpretability, which limits their utility and generalizability across complex systems. Here we propose a novel framework of Interpretable Learning Effective Dynamics (iLED) that offers comparable accuracy to state-of-the-art recurrent neural network-based approaches while providing the added benefit of interpretability. The iLED framework is motivated by Mori-Zwanzig and Koopman operator theory, which justifies the choice of the specific architecture. We demonstrate the effectiveness of the proposed framework in simulations of three benchmark multiscale systems. Our results show that the iLED framework can generate accurate predictions and obtain interpretable dynamics, making it a promising approach for solving high-dimensional multiscale systems.