# Discovering Latent Response Laws in Forced Physical Systems

arXiv: https://arxiv.org/abs/2607.09801 · HTML: https://mlsanity.com/paper/2607.09801/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 51 references that are in the map.
- First posted on arXiv 9 Jul 2026.

## Builds on (most cited first, 28 of 28)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [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)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting](https://mlsanity.com/paper/1912.09363/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Long-term Forecasting with TiDE: Time-series Dense Encoder](https://mlsanity.com/paper/2304.08424/) (2023)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Discovering Sparse Interpretable Dynamics from Partial Observations](https://mlsanity.com/paper/2107.10879/) (2021)
- [Deep Koopman Operator with Control for Nonlinear Systems](https://mlsanity.com/paper/2202.08004/) (2022)
- [Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants](https://mlsanity.com/paper/2211.10575/) (2022)
- [Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2501.13329/) (2025)
- [Discovering physical laws with parallel symbolic enumeration](https://mlsanity.com/paper/2407.04405/) (2024)
- [Continuous-time identification of dynamic state-space models by deep subspace encoding](https://mlsanity.com/paper/2204.09405/) (2022)
- [Distilling Governing Laws and Source Input for Dynamical Systems from Videos](https://mlsanity.com/paper/2205.01314/) (2022)
- [Pixel2Phys: Distilling Governing Laws from Visual Dynamics](https://mlsanity.com/paper/2602.19516/) (2026)

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

- [Seismic Site Response Prediction from Sparse Observations Using Finite-Element-Pretrained Latent Dynamics](https://mlsanity.com/paper/2609.20451/) (2026)

## Abstract (arXiv)

Governing equations provide compact descriptions of physical systems, yet the variables in which they are simple are often hidden in high-dimensional measurements. This challenge is sharper for forced systems, whose responses depend on both intrinsic dynamics and time-dependent inputs. Here we introduce FLARE, a forced latent autoencoder for response equations that learns compact response coordinates, identifies sparse input-dependent latent dynamics and decodes equation rollouts to full responses. By estimating latent dimension from data and separating state estimation from external forcing, FLARE enables forecasts to be initialized from past responses and driven by prescribed future inputs. Across known dynamical systems, application-scale forced responses and visual observations, FLARE recovers compact forced dynamics and predicts long-horizon high-dimensional responses under inputs not used for training. By turning learned coordinates into a dynamical interface, FLARE extends equation discovery to systems whose effective states are hidden within complex observations, providing a route for interpretable modelling and prediction of high-dimensional responses in forced dynamical systems.