Title: SPLIT-PINN: A Separable Probability Learning Approach Using Physics-Informed Neural Networks for High-Dimensional Probabilistic Modeling

Abstract:

This study introduces a probabilistic modeling framework designed to integrate small-scale spatial heterogeneity into macroscopic descriptions of material behavior within polycrystalline metallic systems. By employing probability density functions (PDFs) to represent spatially heterogeneous material state fields, the approach offers a rigorous statistical characterization of microstructural variability and state evolution across various computational polycrystalline realizations. The core of this framework relies on the inverse identification of a probabilistic transport model, which is expressed as a Liouville equation featuring an unknown drift term.

To achieve accurate, stable, and interpretable inference of this drift field in high-dimensional, transport-dominated environments, we propose the Separable Probability Learning Technique via Physics-Informed Neural Networks (SPLIT-PINN). This technique enhances well-posedness, numerical stability, and physical consistency by integrating residual-based adaptive training, orthogonality constraints, and a marginal-correction drift decomposition. Notably, it avoids the need for restrictive parametric assumptions. SPLIT-PINN allows for the direct inference of the drift field, which governs the temporal evolution of joint state PDFs, directly from data.

Following benchmark validation, the framework was applied to physical computational datasets detailing the evolution of polycrystalline microstructural states, specifically focusing on von Mises stress, dislocation density, and equivalent plastic strain rate. The Liouville model, trained on a single dataset, was then utilized for forward predictions of the temporal evolution of both joint and marginal PDFs across multiple unseen polycrystal realizations. Quantitative evaluations against reference PDFs confirm that the proposed framework delivers accurate, robust probabilistic predictions and demonstrates effective generalization across different datasets.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC