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arxiv: 2606.04000 · v1 · pith:B4BDRWWZnew · submitted 2026-05-23 · ❄️ cond-mat.mtrl-sci · cs.LG

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

Pouria Behnoudfar , Deekshith Naidu Ponnana , Noah J. Schmelzer , Janith Wanni , George T. Gray III , Dan J. Thoma , Curt A. Bronkhorst , Nan Chen
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Wenxiao Pan
This is my paper

Pith reviewed 2026-06-30 12:53 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cs.LG
keywords probabilistic modelingphysics-informed neural networksLiouville equationpolycrystalline materialsprobability density functionsdrift field inferencemicrostructural evolutionhigh-dimensional transport
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The pith

A neural network infers an unknown drift field in a Liouville equation from data to predict how probability distributions of material states evolve in polycrystals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to represent small-scale spatial variations in polycrystalline metals through probability density functions rather than single deterministic fields. It formulates the evolution of these PDFs as a Liouville transport equation whose drift term is unknown and must be recovered from simulation data. SPLIT-PINN achieves this recovery by combining a marginal-correction decomposition of the drift, orthogonality constraints, and adaptive residual training, all without assuming a fixed parametric form for the drift. Once trained on one polycrystalline dataset, the resulting model is shown to forecast the time evolution of both joint and marginal PDFs on multiple new, unseen realizations.

Core claim

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.

What carries the argument

SPLIT-PINN, which recovers the drift term of a high-dimensional Liouville equation via marginal-correction drift decomposition, orthogonality constraints, and residual-based adaptive training inside a physics-informed neural network.

If this is right

  • The inferred drift field produces accurate forward predictions of PDF evolution on polycrystal realizations never seen during training.
  • Joint and marginal PDFs of von Mises stress, dislocation density, and plastic strain rate can be forecasted from a single training dataset.
  • The framework supplies a statistical description of microstructural state variability that can be inserted into larger-scale material models.
  • No restrictive parametric form for the drift is required; the neural network learns it directly from data while respecting the transport equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition and training strategy could be tested on other high-dimensional conservation laws that lack closed-form drift expressions.
  • If the learned drift remains stable under modest changes in initial PDF shape, the approach might serve as a surrogate for ensemble averaging in uncertainty-quantification studies.
  • Extension to experimental time-series data would require only that the observed state fields can be cast as evolving PDFs.
  • The orthogonality constraints may also regularize related inverse problems in which a transport velocity must be recovered from noisy density measurements.

Load-bearing premise

The combination of marginal-correction drift decomposition, orthogonality constraints, and residual-based adaptive training is sufficient to make the inverse recovery of the drift field well-posed, stable, and physically consistent in high-dimensional transport problems.

What would settle it

Run the trained model on a new set of polycrystal realizations and measure whether the predicted joint and marginal PDFs deviate systematically from the reference PDFs computed directly from those realizations.

Figures

Figures reproduced from arXiv: 2606.04000 by Curt A. Bronkhorst, Dan J. Thoma, Deekshith Naidu Ponnana, George T. Gray III, Janith Wanni, Nan Chen, Noah J. Schmelzer, Pouria Behnoudfar, Wenxiao Pan.

Figure 1
Figure 1. Figure 1: Overview of the SPLIT-PINN framework. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Benchmark problem: Comparison of the predicted PDF P(x, y, t) against the ground truth. The reported error corresponds to the absolute pointwise error. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Benchmark problem: Recovered drift vector fields. Black: true drift; red: predicted drift. The generic PINN exhibits significant errors, whereas the SPLIT-PINN achieves markedly improved accuracy across the domain. 5. Probabilistic Modeling of Polycrystal State Evolution After validating the effectiveness of SPLIT-PINN on a benchmark problem, we next assess its performance in inferring a predictive statist… view at source ↗
Figure 4
Figure 4. Figure 4: Single crystal model used with a 1000 grain polycrystal simulation in comparison with results of simple compression experiments conducted on polycrystalline pure iron for varying initial temperatures and applied strain rates. The broken lines represent the numerical simulation results. The performance of the single crystal model representing polycrystalline pure iron simple compression experiments is demon… view at source ↗
Figure 5
Figure 5. Figure 5: Five polycrystal statistical volume elements of a quenched and tempered low carbon steel plate. Each numerical model contains 100 × 100 × 100 hexahedral elements with dimensions of 50 µm on a side. Each colored region represents a different single crystal. The colorization scheme is random to enable differentiation of different grains. For each SVE realization, each grain is assigned a specified crystallog… view at source ↗
Figure 6
Figure 6. Figure 6: Equal-area pole figures for measured and reconstructed crystallographic texture using 9,501 orientations. These orientations are assigned randomly to grains in each statistical volume element model displayed in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Location within the shear zone of a forced shear sample simulation (Lieou et al. (2019) of a steel material from which the time evolution of the stress tensor was extracted up to the time at which the simulation indicated adiabatic shear band initiation [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Traction vs. time profiles extracted from the computational element in the forced shear simulation of (Lieou et al. (2019)) are shown by the solid lines. The broken lines are the traction profiles applied to the faces of each statistical volume element in [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Deformed polycrystal cube at the final simulation time of 19.77 µs. The surface displays the contour of von Mises stress in units of MPa. von Mises stress is provided in [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Deformed cross-sectional von Mises stress contour plots of each of the five polycrystal realizations in units of MPa. Each image was taken from the simulation at a time of 19.77 µs [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Cumulative probability distribution function for von Mises stress for all integration points within all five SVE simulations at a time of 19.77 µs. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The joint PDF evolution of dataset D1 used for training SPLIT-PINN at three representative time instants: t = 9.88 µs, t ≈ 12.19 µs, and t = 19.77 µs. Each row corresponds to a 2D cross-section of the joint PDF fixing one state variable (from top to bottom rows: the PDFs with dislocation density=0.18, equivalent plastic strain rate =0.07, and von Mises stress= 0.1, respectively). Time is reported in milli… view at source ↗
Figure 13
Figure 13. Figure 13: Adaptive redistribution of collocation points guided by the marginal PDFs of the von Mises stress and dislocation density. RAD strategy concentrates points in regions of high probability and sharp gradients, improving the resolution of the evolving statistical structure in space and time. Then, we learn the correction drift term to recover the full three-dimensional (3D) drift field. The resulting 3D Liou… view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of the reference and predicted 3D joint PDFs for Dataset D2 at times 13.40, 16.75, and 19.77 µs. The predicted PDFs are obtained by forward propagation using the Liouville model along with the drift field inferred from Dataset D1 [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Absolute pointwise error between the predicted and reference 3D joint PDFs for Dataset D2 at times 13.40, 16.75, and 19.77 µs. The bounded error values indicate that the learned drift field can effectively predict the evolution of the PDF for new polycrystal instances. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: 2D cross-sectional views of the predicted joint PDF for Dataset D2 at times 13.40, 16.75, and 19.77 µs along with the corresponding relative pointwise error compared to the reference PDFs. The results highlight that the learned drift field along with the Liouville model accurately captures the main features of the PDF. Following a similar approach, we extend the generality of the SPLIT-PINN framework by p… view at source ↗
Figure 17
Figure 17. Figure 17: 2D cross-sectional slices of the predicted joint PDF for Dataset D3 at times 13.40, 16.75, and 19.77 µs, together with the relative pointwise error with respect to the reference PDFs. The predicted PDFs accurately reproduce the key probabilistic features, while the largest discrepancies remain confined. In addition to pointwise errors, we quantify the discrepancy between predicted and reference joint PDFs… view at source ↗
Figure 18
Figure 18. Figure 18: 2D cross-sectional slices of the predicted joint PDF for Dataset D4 at times 13.40, 16.75, and 19.77 µs, together with the relative pointwise error compared to the reference PDFs. The results demonstrate that the predicted PDFs capture the dominant probabilistic microstructural state evolution for different polycrystal realizations. Remark 4. The error in the SPLIT-PINN’s results is consistent and overper… view at source ↗
Figure 19
Figure 19. Figure 19: Normalized pointwise KL divergence contours for datasets D2 − D4 , respectively. The uniformly low magnitude demonstrates small discrepancies at both the distribution modes and tails. To verify the conservation property of SPLIT-PINN, we evaluate the total probability M(t) = R Ω P(x, t)dx over time for datasets D2 through D4 , where a value of M ≈ 1 indicates that the predicted joint PDF integrates to uni… view at source ↗
Figure 20
Figure 20. Figure 20: The total probability M(t) = R Ω P(x, t)dx plotted over time for datasets D2 , D3 , and D4 . Values remain close to unity throughout the time range considered, confirming that the predicted PDF preserves total probability. 5.6. Comparison of marginal PDFs While the joint 3D PDFs provide a complete description of the system’s probabilistic state, examining the marginal distributions of each state variable … view at source ↗
Figure 21
Figure 21. Figure 21: Temporal evolution of one-dimensional marginal PDFs for Datasets D1 and D2 . Each subfigure contains the three marginal PDFs obtained by integrating the full PDF over the remaining two dimensions. Results from SPLIT-PINN closely follow the corresponding reference PDFs, illustrating the method’s ability to reproduce the underlying probabilistic dynamics across all datasets. Time is reported in milliseconds… view at source ↗
Figure 22
Figure 22. Figure 22: Temporal evolution of one-dimensional marginal PDFs for Datasets D3 and D4 . Similar to [PITH_FULL_IMAGE:figures/full_fig_p028_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Comparison of the first two moments of the predicted PDFs for von Mises stress across the three datasets (from top: Datasets D2 , D3 , and D4 ). Each subplot illustrates the temporal evolution of the moments (reported in µs), highlighting the model’s ability to reproduce consistent statistical trends across various microstructural conditions [PITH_FULL_IMAGE:figures/full_fig_p029_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Comparison of the first two moments of the predicted PDFs for dislocation density on Datasets D2 , D3 , and D4 . 29 [PITH_FULL_IMAGE:figures/full_fig_p029_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Comparison of the first two moments of the predicted PDFs for equivalent plastic strain rate on Datasets D2 , D3 , and D4 [PITH_FULL_IMAGE:figures/full_fig_p030_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Relative errors on the moments of the predicted PDF compared to the reference ones. From top: Datasets D2 , D3 , and D4 . Time is reported in milliseconds µs. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_26.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces SPLIT-PINN, a physics-informed neural network framework for inverse identification of an unknown drift term in a Liouville equation governing the evolution of joint probability density functions (PDFs) of microstructural state variables (von Mises stress, dislocation density, equivalent plastic strain rate) in polycrystalline materials. The approach employs a marginal-correction drift decomposition, orthogonality constraints, and residual-based adaptive training to infer the drift from a single training dataset without restrictive parametric forms; the resulting model is then applied to forward prediction of joint and marginal PDF evolution on multiple unseen polycrystal realizations, with claims of accurate, robust, and generalizable performance after benchmark validation.

Significance. If the central claims hold, the work would offer a non-parametric route to data-driven probabilistic transport models for high-dimensional microstructural evolution, potentially enabling more reliable incorporation of spatial heterogeneity into macroscopic material descriptions. The provision of machine-checked elements or reproducible code is not mentioned.

major comments (2)
  1. [Abstract / Method description] The central claim of accurate forward PDF predictions on unseen realizations after single-dataset training rests on the inverse problem for the drift being rendered well-posed and stable by the marginal-correction decomposition plus orthogonality constraints. No theorem, condition-number bound, or systematic ablation is supplied to demonstrate that these additions suppress spurious modes or restore uniqueness when the transport term dominates in high dimensions (classically an ill-posed setting). This is load-bearing for the generalization result.
  2. [Abstract] The abstract asserts quantitative agreement with reference PDFs and effective generalization across datasets, yet supplies no error metrics, data-exclusion rules, implementation details, or explicit form of the Liouville equation and decomposition. Without these, the soundness of the physical-consistency claims cannot be verified.
minor comments (2)
  1. [Abstract] The abstract is dense and would benefit from a concise statement of the Liouville equation and the precise form of the marginal-correction decomposition.
  2. [Introduction] Notation for joint versus marginal PDFs and for the drift field should be introduced with explicit symbols early in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable feedback on our manuscript. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [Abstract / Method description] The central claim of accurate forward PDF predictions on unseen realizations after single-dataset training rests on the inverse problem for the drift being rendered well-posed and stable by the marginal-correction decomposition plus orthogonality constraints. No theorem, condition-number bound, or systematic ablation is supplied to demonstrate that these additions suppress spurious modes or restore uniqueness when the transport term dominates in high dimensions (classically an ill-posed setting). This is load-bearing for the generalization result.

    Authors: We agree that a formal theorem establishing well-posedness would strengthen the theoretical foundation. The manuscript instead demonstrates stability and generalization through extensive numerical benchmarks and forward predictions on multiple unseen realizations, showing that the marginal-correction decomposition combined with orthogonality constraints and adaptive training yields consistent results without parametric assumptions. We will add a dedicated discussion subsection with additional ablation experiments quantifying the contribution of each constraint to suppressing non-physical modes. revision: partial

  2. Referee: [Abstract] The abstract asserts quantitative agreement with reference PDFs and effective generalization across datasets, yet supplies no error metrics, data-exclusion rules, implementation details, or explicit form of the Liouville equation and decomposition. Without these, the soundness of the physical-consistency claims cannot be verified.

    Authors: The abstract provides a concise overview, while the explicit Liouville equation, decomposition, error metrics, and implementation details appear in Sections 2–4. To improve standalone readability we will revise the abstract to include representative quantitative error values and a brief statement of the equation form, respecting length constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected.

full rationale

The provided abstract and context describe a standard data-driven workflow: the drift term in the Liouville equation is inferred from training data via SPLIT-PINN (incorporating marginal-correction decomposition, orthogonality constraints, and residual-adaptive training), after which the resulting model is applied to forward evolution on unseen polycrystal realizations with quantitative comparison to reference PDFs. This does not reduce to a self-definitional loop, fitted-input-renamed-as-prediction, or self-citation chain by the paper's own statements. No equations or citations are supplied that would exhibit the central claim being equivalent to its inputs by construction. The emphasis on generalization across datasets indicates an independent test rather than tautology. The well-posedness assumption is presented as an enabling technique but is not shown to be circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on the domain assumption that the Liouville equation is the correct transport model and that the listed enhancements suffice for well-posed inference. No explicit free parameters or invented physical entities are named.

axioms (1)
  • domain assumption The evolution of joint state PDFs is governed by a Liouville equation with unknown drift term
    Framework is built on inverse identification of a probabilistic transport model formulated as a Liouville equation.

pith-pipeline@v0.9.1-grok · 5839 in / 1256 out tokens · 41632 ms · 2026-06-30T12:53:45.981892+00:00 · methodology

discussion (0)

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