# mLaSDI: Multi-stage latent space dynamics identification

arXiv: https://arxiv.org/abs/2506.09207 · HTML: https://mlsanity.com/paper/2506.09207/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 58 references that are in the map.
- First posted on arXiv 10 Jun 2025.

## Builds on (most cited first, 19 of 19)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification](https://mlsanity.com/paper/2204.12005/) (2022)
- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [DR-RNN: A deep residual recurrent neural network for model reduction](https://mlsanity.com/paper/1709.00939/) (2017)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Component-wise reduced order model lattice-type structure design](https://mlsanity.com/paper/2010.10770/) (2020)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [Train Small, Model Big: Scalable Physics Simulators via Reduced Order Modeling and Domain Decomposition](https://mlsanity.com/paper/2401.10245/) (2023)
- [Certified data-driven physics-informed greedy auto-encoder simulator](https://mlsanity.com/paper/2211.13698/) (2022)

## Built on by (most cited first, 2 of 2)

- [Rollout-LaSDI: Enhancing the long-term accuracy of Latent Space Dynamics](https://mlsanity.com/paper/2509.08191/) (2025)
- [Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives](https://mlsanity.com/paper/2512.15997/) (2025)

## Abstract (arXiv)

Accurately solving partial differential equations (PDEs) is essential across many scientific disciplines. However, high-fidelity solvers can be computationally prohibitive, motivating the development of reduced-order models (ROMs). Recently, Latent Space Dynamics Identification (LaSDI) was proposed as a data-driven, non-intrusive ROM framework. LaSDI compresses the training data via an autoencoder and learns user-specified ordinary differential equations (ODEs), governing the latent dynamics, enabling rapid predictions for unseen parameters. While LaSDI has produced effective ROMs for numerous problems, the autoencoder must simultaneously reconstruct the training data and satisfy the imposed latent dynamics, which are often competing objectives that limit accuracy, particularly for complex or high-frequency phenomena. To address this limitation, we propose multi-stage Latent Space Dynamics Identification (mLaSDI). With mLaSDI, we train LaSDI sequentially in stages. After training the initial autoencoder, we train additional decoders which map the latent trajectories to residuals from previous stages. This staged residual learning, combined with periodic activation functions, enables recovery of high-frequency content without sacrificing interpretability of the latent dynamics. We further provide an error decomposition separating autoencoder and latent dynamics contributions, and prove that additional training stages cannot increase the training residual. Numerical experiments on a multiscale oscillating system, unsteady wake flow, and the 1D-1V Vlasov equation demonstrate that mLaSDI achieves significantly lower reconstruction and prediction errors, often by an order of magnitude, while requiring less training time and reduced hyperparameter tuning compared to standard LaSDI.