# Weak-Form Latent Space Dynamics Identification

arXiv: https://arxiv.org/abs/2311.12880 · HTML: https://mlsanity.com/paper/2311.12880/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 53 references that are in the map.
- First posted on arXiv 20 Nov 2023.

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

- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (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)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization](https://mlsanity.com/paper/2206.00176/) (2022)
- [Application of Convolutional Neural Network to Predict Airfoil Lift Coefficient](https://mlsanity.com/paper/1712.10082/) (2017)
- [Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques](https://mlsanity.com/paper/2107.11460/) (2021)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Efficient nonlinear manifold reduced order model](https://mlsanity.com/paper/2011.07727/) (2020)
- [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022)
- [Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population](https://mlsanity.com/paper/2204.14141/) (2022)
- [Certified data-driven physics-informed greedy auto-encoder simulator](https://mlsanity.com/paper/2211.13698/) (2022)

## Built on by (most cited first, 20 of 20)

- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [The Weak Form Is Stronger Than You Think](https://mlsanity.com/paper/2409.06751/) (2024)
- [Reduced-order modeling for parameterized PDEs via implicit neural representations](https://mlsanity.com/paper/2311.16410/) (2023)
- [Thermodynamically Consistent Latent Dynamics Identification for Parametric Systems](https://mlsanity.com/paper/2506.08475/) (2025)
- [Weak Form Scientific Machine Learning: Test Function Construction for System Identification](https://mlsanity.com/paper/2507.03206/) (2025)
- [Latent Twins](https://mlsanity.com/paper/2509.20615/) (2025)
- [mLaSDI: Multi-stage latent space dynamics identification](https://mlsanity.com/paper/2506.09207/) (2025)
- [Rollout-LaSDI: Enhancing the long-term accuracy of Latent Space Dynamics](https://mlsanity.com/paper/2509.08191/) (2025)
- [Sequential decoder training for improved latent space dynamics identification](https://mlsanity.com/paper/2510.03535/) (2025)
- [Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives](https://mlsanity.com/paper/2512.15997/) (2025)
- [Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy](https://mlsanity.com/paper/2605.30432/) (2026)
- [Understanding the role of autoencoders for stiff dynamical systems using information theory](https://mlsanity.com/paper/2503.06325/) (2025)
- [Latent Space Element Method](https://mlsanity.com/paper/2601.01741/) (2026)
- [Data-Driven Model Reduction using WeldNet: Windowed Encoders for Learning Dynamics](https://mlsanity.com/paper/2512.11090/) (2025)
- [Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamics](https://mlsanity.com/paper/2603.05201/) (2026)
- [WGFINNs: Weak formulation-based GENERIC formalism informed neural networks](https://mlsanity.com/paper/2604.02601/) (2026)
- [Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamics](https://mlsanity.com/paper/2605.20639/) (2026)
- [Physics-conforming Latent Twins](https://mlsanity.com/paper/2606.15053/) (2026)
- [Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamics](https://mlsanity.com/paper/2609.09434/) (2026)

## Abstract (arXiv)

Recent work in data-driven modeling has demonstrated that a weak formulation of model equations enhances the noise robustness of a wide range of computational methods. In this paper, we demonstrate the power of the weak form to enhance the LaSDI (Latent Space Dynamics Identification) algorithm, a recently developed data-driven reduced order modeling technique. We introduce a weak form-based version WLaSDI (Weak-form Latent Space Dynamics Identification). WLaSDI first compresses data, then projects onto the test functions and learns the local latent space models. Notably, WLaSDI demonstrates significantly enhanced robustness to noise. With WLaSDI, the local latent space is obtained using weak-form equation learning techniques. Compared to the standard sparse identification of nonlinear dynamics (SINDy) used in LaSDI, the variance reduction of the weak form guarantees a robust and precise latent space recovery, hence allowing for a fast, robust, and accurate simulation. We demonstrate the efficacy of WLaSDI vs. LaSDI on several common benchmark examples including viscid and inviscid Burgers', radial advection, and heat conduction. For instance, in the case of 1D inviscid Burgers' simulations with the addition of up to 100% Gaussian white noise, the relative error remains consistently below 6% for WLaSDI, while it can exceed 10,000% for LaSDI. Similarly, for radial advection simulations, the relative errors stay below 15% for WLaSDI, in stark contrast to the potential errors of up to 10,000% with LaSDI. Moreover, speedups of several orders of magnitude can be obtained with WLaSDI. For example applying WLaSDI to 1D Burgers' yields a 140X speedup compared to the corresponding full order model. Python code to reproduce the results in this work is available at (https://github.com/MathBioCU/PyWSINDy_ODE) and (https://github.com/MathBioCU/PyWLaSDI).