# Symbolic Pregression: Discovering Physical Laws from Distorted Video

arXiv: https://arxiv.org/abs/2005.11212 · HTML: https://mlsanity.com/paper/2005.11212/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 May 2020.

## Built on by (most cited first, 21 of 21)

- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [ODEFormer: Symbolic Regression of Dynamical Systems with Transformers](https://mlsanity.com/paper/2310.05573/) (2023)
- [Learning to Accelerate Partial Differential Equations via Latent Global Evolution](https://mlsanity.com/paper/2206.07681/) (2022)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [Grokking as Compression: A Nonlinear Complexity Perspective](https://mlsanity.com/paper/2310.05918/) (2023)
- [Machine-Learning Non-Conservative Dynamics for New-Physics Detection](https://mlsanity.com/paper/2106.00026/) (2021)
- [SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Vision](https://mlsanity.com/paper/2111.05986/) (2021)
- [Similarity-Based Equational Inference in Physics](https://mlsanity.com/paper/2103.13496/) (2021)
- [Bayesian Learning to Discover Mathematical Operations in Governing Equations of Dynamic Systems](https://mlsanity.com/paper/2206.00669/) (2022)
- [Active Learning in Symbolic Regression with Physical Constraints](https://mlsanity.com/paper/2305.10379/) (2023)
- [Analyzing Generalization in Pre-Trained Symbolic Regression](https://mlsanity.com/paper/2509.19849/) (2025)
- [Distilling Governing Laws and Source Input for Dynamical Systems from Videos](https://mlsanity.com/paper/2205.01314/) (2022)
- [Symbolic Metamodels for Interpreting Black-boxes Using Primitive Functions](https://mlsanity.com/paper/2302.04791/) (2023)
- [A Survey on Video Analytics in Cloud-Edge-Terminal Collaborative Systems](https://mlsanity.com/paper/2502.06581/) (2025)
- [Uncovering Closed-form Governing Equations of Nonlinear Dynamics from Videos](https://mlsanity.com/paper/2106.04776/) (2021)
- [Microscopy is All You Need](https://mlsanity.com/paper/2210.06526/) (2022)
- [Quantifying Complexity: An Object-Relations Approach to Complex Systems](https://mlsanity.com/paper/2210.12347/) (2022)
- [Vision-based Discovery of Nonlinear Dynamics for 3D Moving Target](https://mlsanity.com/paper/2404.17865/) (2024)
- [Optimization of geological carbon storage operations with multimodal latent dynamic model and deep reinforcement learning](https://mlsanity.com/paper/2406.04575/) (2024)
- [Symbolic regression for defect interactions in 2D materials](https://mlsanity.com/paper/2512.20785/) (2025)

## Abstract (arXiv)

We present a method for unsupervised learning of equations of motion for objects in raw and optionally distorted unlabeled video. We first train an autoencoder that maps each video frame into a low-dimensional latent space where the laws of motion are as simple as possible, by minimizing a combination of non-linearity, acceleration and prediction error. Differential equations describing the motion are then discovered using Pareto-optimal symbolic regression. We find that our pre-regression ("pregression") step is able to rediscover Cartesian coordinates of unlabeled moving objects even when the video is distorted by a generalized lens. Using intuition from multidimensional knot-theory, we find that the pregression step is facilitated by first adding extra latent space dimensions to avoid topological problems during training and then removing these extra dimensions via principal component analysis.