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

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.