# Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamics

arXiv: https://arxiv.org/abs/2609.09434 · HTML: https://mlsanity.com/paper/2609.09434/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 56 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data](https://mlsanity.com/paper/2009.08810/) (2020)
- [Numerical differentiation of noisy data: A unifying multi-objective optimization framework](https://mlsanity.com/paper/2009.01911/) (2020)
- [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data](https://mlsanity.com/paper/2001.04816/) (2020)
- [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [The Weak Form Is Stronger Than You Think](https://mlsanity.com/paper/2409.06751/) (2024)
- [A practical identifiability criterion leveraging weak-form parameter estimation](https://mlsanity.com/paper/2506.17373/) (2025)
- [Weak Form Scientific Machine Learning: Test Function Construction for System Identification](https://mlsanity.com/paper/2507.03206/) (2025)
- [Learning Physically Interpretable Atmospheric Models from Data with WSINDy](https://mlsanity.com/paper/2501.00738/) (2025)
- [Bias and Coverage Properties of the WENDy-IRLS Algorithm](https://mlsanity.com/paper/2510.03365/) (2025)
- [Weak Form Learning for Mean-Field Partial Differential Equations: an Application to Insect Movement](https://mlsanity.com/paper/2510.07786/) (2025)
- [Physics-guided weak-form discovery of reduced-order models for trapped ultracold hydrodynamics](https://mlsanity.com/paper/2406.07519/) (2024)
- [WENDy for Nonlinear-in-Parameters ODEs](https://mlsanity.com/paper/2502.08881/) (2025)
- [Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy](https://mlsanity.com/paper/2605.30432/) (2026)
- [WSINDy for Model Predictive Control with Applications to Fusion, Drones, and Chaos](https://mlsanity.com/paper/2604.23269/) (2026)
- [Robust Moment-Based Estimation via Spectral Gradient Reweighting](https://mlsanity.com/paper/2605.27718/) (2026)

## Abstract (arXiv)

Weak Sparse Identification of Nonlinear Dynamics (WSINDy) provides a noise-robust approach for learning dynamical systems from data without requiring numerical differentiation. However, for high-dimensional systems, tensor-product libraries of candidate functions grow exponentially with the state dimension, making standard WSINDy expensive in both computation and memory. The Multidimensional Approximation of Nonlinear Dynamics (MANDy) addresses this scaling through a tensor-train (TT) representation of the candidate library, but does not provide a mechanism for sparse model selection. Here, we combine these approaches to develop TT-WSINDy, which performs the weak-form transformation, regression, and sparsification in TT format. We show that the TT formulation recovers the corresponding WSINDy regression problem and derive polynomial time and memory complexity bounds for the tensor-train sparsification procedure. Numerical experiments demonstrate robustness to measurement noise and computational savings for high-dimensional systems.