# SPLIT-PINN: Separable Probability Learning Technique via Physics-Informed Neural Networks for High-Dimensional Probabilistic Modeling

arXiv: https://arxiv.org/abs/2606.04000 · HTML: https://mlsanity.com/paper/2606.04000/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 71 references that are in the map.
- First posted on arXiv 23 May 2026.

## Builds on (most cited first, 15 of 15)

- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity](https://mlsanity.com/paper/2203.08802/) (2022)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Understanding and Mitigating Extrapolation Failures in Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.09478/) (2023)
- [Learning dislocation dynamics mobility laws from large-scale MD simulations](https://mlsanity.com/paper/2309.14450/) (2023)
- [PC-SRGAN: Physically Consistent Super-Resolution Generative Adversarial Network for General Transient Simulations](https://mlsanity.com/paper/2505.06502/) (2025)

## Abstract (arXiv)

We present a probabilistic modeling framework for incorporating small-scale spatial heterogeneity into macroscopic descriptions of material behavior for polycrystalline metallic materials. Spatially heterogeneous material state fields are represented using probability density functions (PDFs), providing a principled statistical description of microstructural variability and state evolution across different computational polycrystalline realizations. The framework is built on the inverse identification of a probabilistic transport model, formulated as a Liouville equation with an unknown drift term. To enable accurate, stable, and interpretable inference of this drift field in high-dimensional, transport-dominated settings, we develop a Separable Probability Learning Technique via Physics-Informed Neural Networks (SPLIT-PINN). This method incorporates a marginal-correction drift decomposition, orthogonality constraints, and residual-based adaptive training to enhance well-posedness, numerical stability, and physical consistency without imposing restrictive parametric assumptions. Using SPLIT-PINN, the drift field governing the temporal evolution of joint state PDFs is inferred directly from data. After benchmark validation, the framework is applied to physical computational datasets describing the evolution of polycrystalline microstructural states, including von Mises stress, dislocation density, and equivalent plastic strain rate. The learned Liouville model, trained on a single dataset, is subsequently used in forward predictions of the temporal evolution of joint and marginal PDFs for multiple unseen polycrystal realizations. Quantitative comparisons with reference PDFs demonstrate that the proposed framework yields accurate and robust probabilistic predictions and generalizes effectively across datasets.