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

Akshay Dhalpe; Arnab Chattopadhyay; Shahriar Bakrani Balani; Eric Cortanéa; Sasu Tarkoma

cs.LGOct 12, 2026

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

Authors: Akshay Dhalpe, Arnab Chattopadhyay, Shahriar Bakrani Balani, Eric Cortanéa, Sasu Tarkoma

Organizations: Tampere University · University of Helsinki

Abstract

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.

Figures & tables

	Baseline			$\pi$ -Net
Model	$R^{2}$	MAPE	RMSE	$R^{2}$	MAPE	RMSE
LASSO	0.742	10.69%	1.919	0.840	7.92%	1.511
MLP	0.809	7.39%	1.652	0.851	7.14%	1.461
FT-Transformer	0.796	8.14%	1.708	0.864	6.93%	1.394

Table 1: Performance comparison of backbone architectures and their

\pi

-Net variants.

	LASSO		MLP		FT-Transformer
Training split	Baseline	$\pi$ -Net	Baseline	$\pi$ -Net	Baseline	$\pi$ -Net
80% train; 20% test	0.742	0.840	0.809	0.851	0.796	0.864
75% train; 25% test	0.697	0.830	0.832	0.851	0.782	0.852
67% train; 33% test	0.712	0.833	0.799	0.848	0.770	0.840
50% train; 50% test	0.680	0.821	0.792	0.825	0.679	0.825

Table 2: Effect of training set size on the cross-validation

R^{2}

performance of backbone architectures and their

\pi

-Net variants.

Figure 1: Directed graph of the GMAW process showing the directed propagation path from arc voltage ( $U_{a}$ ) to weld bead cross-sectional area (A)

Backbone Model	Architecture Configuration	Key Mechanism & Rationale
LASSO	Single linear layer mapping the 4 input features to 5 dimensionless coefficient outputs ( $\pi_{2}-\pi_{6}$ ).	Employs an $L_{1}$ penalty to perform feature selection, identifying which welding parameters exert the greatest influence on each dimensionless group.
MLP	Input: 4 neurons Hidden: 2 layers (16 neurons each) Output: 5 neurons corresponding to the coefficients $\pi_{2}-\pi_{6}$	Compact design deliberately limits trainable parameters to mitigate overfitting risks on the small experimental dataset ( $n=60$ ). Aligns with standard, robust baselines for GMAW [47,48].
FT-Transformer	Tokenization: Independent linear tokenizer projects each feature to a 64-dimensional embedding with additive learnable positional embeddings. Encoder: A CLS token is prepended (forming a 5-token sequence), processed by a 2-layer pre-norm transformer encoder. Output: Learned CLS representation is decoded to $\pi_{2}-\pi_{6}$ .	Attention mechanism explicitly models the complex, strong interdependencies between welding parameters (e.g., arc voltage and wire feed rate).

Table 3: Architectural configuration and structural rationale for the evaluated ML backbones.

Physics-Informed ML

Physics-informed machine learning combines data-driven models with physical laws to solve, predict, and infer scientific systems.

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pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learning

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School of Mechanical Engineering, Purdue University, West Lafayette, 47906, IN, USA. · Department of Mathematics, Purdue University, West Lafayette, 47906, IN, USA.

Physics-Informed Generative Modeling PDE Surrogate Modeling Partial Differential Equations Physics-Informed ML PDE Inverse Problems PDE Discovery

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Generalizing across disparate physical laws remains a fundamental challenge for artificial intelligence in science. Existing deep-learning solvers are largely confined to single-equation settings, limiting transfer across physical regimes and inference tasks. Here we introduce pADAM, a unified generative framework that learns a shared probabilistic prior across heterogeneous partial differential equation families. Through a learned joint distribution of system states and, where applicable, physical parameters, pADAM supports forward prediction and inverse inference within a single architecture without retraining. Across benchmarks ranging from scalar diffusion to nonlinear Navier--Stokes equations, pADAM achieves accurate inference even under sparse observations. Combined with conformal prediction, it also provides reliable uncertainty quantification with coverage guarantees. In addition, pADAM performs probabilistic model selection from only two sparse snapshots, identifying governing laws through its learned generative representation. These results highlight the potential of generative multi-physics modeling for unified and uncertainty-aware scientific inference.

Abstract

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