# pi-Net: Empirical Latent Representations for Differentiable Physics Heads in Data-Scarce Systems

arXiv: https://arxiv.org/abs/2610.16504 · HTML: https://mlsanity.com/paper/2610.16504/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 9 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 4 of 4)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Dimensionally Consistent Learning with Buckingham Pi](https://mlsanity.com/paper/2202.04643/) (2022)
- [Toward Knowledge-Guided AI for Inverse Design in Manufacturing: A Perspective on Domain, Physics, and Human-AI Synergy](https://mlsanity.com/paper/2506.00056/) (2025)

## Abstract (arXiv)

Developing reliable predictive models for multi-physics systems often forces a compromise between computationally demanding physical simulators and data-hungry, opaque machine learning (ML) algorithms. While incorporating dimensional analysis into directed computation graphs offers a structurally sound alternative to explicitly solving partial differential equations (PDEs), such formulations typically yield underdetermined systems that act as imperfect simulators. We introduce pi-Net, a physics-informed latent variable model that resolves these imperfect simulators. The hybrid pi-Net architecture consists of an encoder that maps inputs into physically normalized latent space, which are then decoded into final predictions by a differentiable physics head. Evaluated on a real-world, data-scarce Gas Metal Arc Welding (GMAW) case study, the framework achieves an R2 of 0.864 with only 60 experimental samples. The results demonstrate that pi-Net significantly outperforms purely data-driven baselines in data-scarce regimes while providing causal, process-level explainability.