# Physics-Grounded Motion Forecasting via Equation Discovery for Trajectory-Guided Image-to-Video Generation

arXiv: https://arxiv.org/abs/2507.06830 · HTML: https://mlsanity.com/paper/2507.06830/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 69 references that are in the map.
- First posted on arXiv 9 Jul 2025.

## Builds on (most cited first, 27 of 27)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Stable Video Diffusion: Scaling Latent Video Diffusion Models to Large Datasets](https://mlsanity.com/paper/2311.15127/) (2023)
- [Towards Accurate Generative Models of Video: A New Metric & Challenges](https://mlsanity.com/paper/1812.01717/) (2018)
- [Cosmos World Foundation Model Platform for Physical AI](https://mlsanity.com/paper/2501.03575/) (2025)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [VideoGPT: Video Generation using VQ-VAE and Transformers](https://mlsanity.com/paper/2104.10157/) (2021)
- [MotionCtrl: A Unified and Flexible Motion Controller for Video Generation](https://mlsanity.com/paper/2312.03641/) (2023)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [StableVideo: Text-driven Consistency-aware Diffusion Video Editing](https://mlsanity.com/paper/2308.09592/) (2023)
- [AMT: All-Pairs Multi-Field Transforms for Efficient Frame Interpolation](https://mlsanity.com/paper/2304.09790/) (2023)
- [Tora: Trajectory-oriented Diffusion Transformer for Video Generation](https://mlsanity.com/paper/2407.21705/) (2024)
- [DragAnything: Motion Control for Anything using Entity Representation](https://mlsanity.com/paper/2403.07420/) (2024)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [WISA: World Simulator Assistant for Physics-Aware Text-to-Video Generation](https://mlsanity.com/paper/2503.08153/) (2025)
- [Symbolic Regression with a Learned Concept Library](https://mlsanity.com/paper/2409.09359/) (2024)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [3DTrajMaster: Mastering 3D Trajectory for Multi-Entity Motion in Video Generation](https://mlsanity.com/paper/2412.07759/) (2024)
- [SG-I2V: Self-Guided Trajectory Control in Image-to-Video Generation](https://mlsanity.com/paper/2411.04989/) (2024)
- [MotionCanvas: Cinematic Shot Design with Controllable Image-to-Video Generation](https://mlsanity.com/paper/2502.04299/) (2025)
- [SNIP: Bridging Mathematical Symbolic and Numeric Realms with Unified Pre-training](https://mlsanity.com/paper/2310.02227/) (2023)
- [StoryGPT-V: Large Language Models as Consistent Story Visualizers](https://mlsanity.com/paper/2312.02252/) (2023)
- [CausalScore: An Automatic Reference-Free Metric for Assessing Response Relevance in Open-Domain Dialogue Systems](https://mlsanity.com/paper/2406.17300/) (2024)
- [Uncovering Closed-form Governing Equations of Nonlinear Dynamics from Videos](https://mlsanity.com/paper/2106.04776/) (2021)

## Built on by (most cited first, 4 of 4)

- [NewtonGen: Physics-Consistent and Controllable Text-to-Video Generation via Neural Newtonian Dynamics](https://mlsanity.com/paper/2509.21309/) (2025)
- [SeeU: Seeing the Unseen World via 4D Dynamics-aware Generation](https://mlsanity.com/paper/2512.03350/) (2025)
- [NEXUS: Neural Energy Fields for Physically Consistent Contact-Rich 3D Object Dynamics](https://mlsanity.com/paper/2606.15015/) (2026)
- [Phase-aware video generation for physics-grounded dynamics and interactions](https://mlsanity.com/paper/2610.11791/) (2026)

## Abstract (arXiv)

Recent advances in diffusion-based and autoregressive video generation models have achieved remarkable visual realism. However, these models typically lack accurate physical alignment, failing to replicate real-world dynamics in object motion. This limitation arises primarily from their reliance on learned statistical correlations rather than capturing mechanisms adhering to physical laws. To address this issue, we introduce a novel framework that integrates symbolic regression (SR) and trajectory-guided image-to-video (I2V) models for physics-grounded video forecasting. Our approach extracts motion trajectories from input videos, uses a retrieval-based pre-training mechanism to enhance symbolic regression, and discovers equations of motion to forecast physically accurate future trajectories. These trajectories then guide video generation without requiring fine-tuning of existing models. Evaluated on scenarios in Classical Mechanics, including spring-mass, pendulums, and projectile motions, our method successfully recovers ground-truth analytical equations and improves the physical alignment of generated videos over baseline methods.