A physics-aware deep learning model for shear band formation around collapsing pores in shocked reactive materials

AI summary

Overview

Research area: Physics-informed machine learning applied to solid mechanics and shock physics of energetic materials (energetic crystals such as RDX and HMX).

Technical level: Advanced. The paper assumes familiarity with partial differential equations, numerical integration schemes (Runge–Kutta), convolutional neural networks, neural operators, and continuum shock physics.

Scope: The paper evaluates an improved physics-aware deep learning architecture (PARCv2) against two other physics-informed machine learning models for predicting shear band formation and plastic heating around a collapsing pore in shocked RDX under weak-to-moderate shock loading.

What This Paper Is About

Shock initiation of energetic materials produces "hotspots" — localized regions of intense energy concentration that can trigger chemical energy release and detonation. While hotspot formation under strong shocks has been well studied, the weak-to-moderate shock regime, where plastic dissipation and shear bands compete with pore collapse, remains underexplored because direct numerical simulations of shear bands are computationally expensive. This paper improves and benchmarks a physics-aware deep learning model to rapidly predict shear band formation and temperature evolution around a collapsing 50 nm pore, with the goal of accelerating the ensembles of simulations needed for microstructure-informed reactive burn models.

Key Contributions

1. An improved PARCv2 architecture for weak-to-moderate shock regimes. The authors replaced the original right-derivative advection calculation with an upwind numerical scheme, extended boundary condition support beyond zero padding to include constant, periodic, and zero-gradient conditions (zero-gradient was used here), and added a multi-step curriculum training strategy. The model contains approximately 15.7 million trainable parameters.

2. A systematic benchmark against two widely used physics-informed models. PARCv2 was compared to a Tensorized Fourier Neural Operator (TFNO2d) and a ResNet-based neural ODE, all trained on the same data split, with the neural ODE containing approximately 15.5 million trainable parameters and FNO using four Fourier blocks, the first 24 Fourier modes, and 78 channels.

3. Identification and analysis of failure modes across all three models. The paper documents spectral bias, boundary condition enforcement issues, and performance deterioration under low impact velocity, breaking the low-velocity degradation into three likely causes.

4. Physics-driven evaluation metrics beyond RMSE. The authors compared temperature and pressure distribution functions at complete pore collapse and extracted dominant shear band location, width, and temperature increase, finding that simpler descriptive metrics can be misleading (for example, at 1200 m/s FNO matched the location metric better than PARCv2 but completely missed the double-peak shear band structure).

Main Findings

• PARCv2 achieves the best overall accuracy. It consistently produced comparable or lower RMSE than the other two models across all five predicted fields (temperature, pressure, microstructure, and X- and Y-velocity), with particularly notable reductions in low-velocity extrapolation cases. Compared to the neural ODE, PARCv2 achieved nearly a tenfold decrease in RMSE across the five fields.

• PARCv2 produces sharper dominant shear bands than FNO. Visual examination at 1800 m/s showed both models captured overall patterns, but PARCv2's dominant shear bands were noticeably sharper and closer to ground truth. At 3000 m/s, PARCv2's RMSE along the shear band profile was approximately 50% lower than FNO's, and PARCv2 was free of nonphysical temperature increases (FNO showed an anomalous peak at y ∼ 110 nm).

• The ResNet-based neural ODE failed to produce reasonable predictions. It is labeled "Fail" in the physics-driven metrics table for the 1200, 1800, and 2400 m/s cases, and failed at 800 m/s as well.

• All models miss weak and fine-scale structures. Temperature RMSE within the training set coverage was 25–50 K, comparable to or sometimes greater than the temperature difference between shear bands and surrounding material, suggesting non-dominant shear bands may be absent from all AI predictions. Peak temperatures within shear bands were underestimated by 10–40 K relative to the surrounding background.

• Spectral bias is evident during training. The model learns large-scale patterns (high-temperature regions, shock waves, reverse ballistics) earlier in training, while smaller-scale patterns (shear bands, shape and asymmetry) are learned later.

• Best accuracy occurred in the 1200 m/s to 2400 m/s range. All models achieved their best prediction accuracy in this intermediate velocity band.

• Low-velocity extrapolation is the hardest failure mode. At 500 m/s, RMSE was approximately twice the lowest error achieved by the models, and relative error in average temperature increased nearly tenfold. FNO's RMSE increase in the temperature, pressure, microstructure, and Y-velocity fields often exceeded tenfold, compared to typically around a twofold increase for PARCv2. Peak temperatures in shear bands were more than 50 K too low and predicted bands were significantly wider than ground truth. All three models were labeled "Fail" at 500 m/s.

• FNO produced numerical artifacts in high-velocity extrapolation. Beyond t = 32.5 ps at 3000 m/s, FNO's predictions became unreliable, while PARCv2 showed no noticeable artifacts, only blurriness and inaccuracies in the central high-temperature region shape.

• PARCv2 systematically underpredicts extremes. In temperature and pressure distribution functions, ParCv2 consistently underpredicts the highest temperatures and maximum pressures, with no predicted pressure values exceeding 100 GPa. It also struggles with the discontinuous gap between 0 K and 200 K present in the ground truth distributions.

• Locations and widths were often accurate even when temperatures were not. For 1800 m/s and 2400 m/s, both PARCv2 and FNO predicted the peak-temperature location with zero error and reproduced dominant shear band width with a one-pixel deviation, despite underpredicting the temperature increase.

• The 2800 m/s case had nearly twice the RMSE of the 800 m/s case but looked visually closer to ground truth. The authors attribute this to the lower absolute temperatures in the 800 m/s case making deviations more noticeable.

Methodology in Plain English

The researchers used existing continuum simulation data (generated with the SCIMITAR3D code, previously published by Herrin et al.) rather than running new physics simulations. Each simulation models a rectangular RDX block containing a single 50 nm circular pore, launched at a rigid wall at velocities from 720 m/s to 2880 m/s. Data were rasterized onto a 128 × 256 grid at 1.1719 nm per pixel and 2.5 ps between snapshots.

The dataset was split into 70 training simulations and 18 validation simulations, with a 26-simulation hold-out test set consisting of velocities divisible by 100 within the training range plus two extrapolation bands (500–700 m/s and 2900–3080 m/s). Temperature and pressure channels were min-max normalized, velocity fields were normalized as vectors between -1 and 1, and normalization constants came only from the training set.

Three models were trained on the same split. PARCv2 embeds the structure of advection-diffusion-reaction equations directly into the network: advection of the five state variables is computed explicitly with an upwind scheme, additional couplings are handled by a U-Net, and temporal integration uses fourth-order Runge–Kutta. It was trained for 3000 epochs predicting one timestep (learning rate 10⁻⁴), then 1000 epochs predicting three timesteps (learning rate 10⁻⁵), using MAE loss, the Adam optimizer, no regularization, and four Nvidia A100 GPUs over 72 hours with PyTorch DDP. FNO learns operators in Fourier space and was trained recursively for 10,000 epochs at learning rate 10⁻⁴. The neural ODE uses a ResNet to predict temporal derivatives, which are then integrated with RK4, and was trained for 3000 epochs at learning rate 10⁻⁴. All models take a single timestep as input but can roll out arbitrarily many steps; test roll-outs ranged from 15 to 50 steps depending on impact velocity.

Evaluation combined RMSE across all five fields as a function of impact velocity, visual inspection of roll-out fields, temperature and pressure distribution functions, and a focused analysis of temperature profiles along a vertical slice at X = 29.30 nm for 29.30 < Y < 175.79 nm.

Why This Matters

Impact on research: The work demonstrates that embedding physics structure into a neural network (explicit advection, U-Net couplings, RK4 integration) yields substantial accuracy gains over purely learned operators for this problem. It also shows that popular physics-informed approaches like FNO and neural ODE have significant, characteristic failure modes in this regime — FNO loses stability in out-of-distribution cases, and neural ODE fails to reproduce shear band patterns at all. The paper argues that domain-specific considerations matter more than generic architectural choices when building AI-accelerated simulation tools, and it quantifies the gap between continuum simulation cost (O(1) CPU hour) and atomistic simulation cost (O(10³) CPU hours) that motivates AI acceleration.

Real-world applications:

• Safe storage and handling of energetic materials, which the paper identifies as a central motivation.
• Transportation and development of energetic materials, where controlling energy release is a safety requirement.
• Accelerating microstructure-informed reactive burn model construction, which requires ensembles of simulations that remain computationally intensive even with continuum codes.
• Providing fast surrogate predictions of hotspot temperature fields for shock-to-detonation and deflagration-to-detonation transition studies.

Industry relevance: Defense and energetic materials industries, propellant and explosives manufacturers, and computational physics software developers all depend on predicting shock initiation behavior. The publicly released code (github.com/baeklab/PARCtorch and github.com/chengxinlun/shear_bands) makes the improved architecture directly reusable, and the identified failure modes give practitioners concrete guidance on where AI surrogates can and cannot be trusted.

Future Directions

• Reducing spectral bias. The paper explicitly asks whether spectral bias reduction methods proven in the computer vision community can translate to physics modeling problems, since the tendency to learn low-frequency structure first directly causes missed weak shear bands.

• Better boundary condition enforcement. The authors examined the theoretical and computational foundations for enforcing various boundary conditions through different padding operations in convolutional neural networks, leaving room for principled padding schemes beyond the zero-gradient condition used here.

• Fixing low-velocity performance degradation. The paper breaks the performance decrease at low impact velocities into three likely reasons, and resolving them is presented as an open challenge — particularly since at 500 m/s all three models failed entirely.

• Extending beyond a single pore. The simulations use one 50 nm pore in RDX; the paper's motivation for building these surrogates is ensembles of microstructure-resolved simulations, which would require handling many pores, defects, and heterogeneous microstructures.

• Handling the strong-shock regime transition. The 3000 m/s case could not be assessed with the shear band metrics because shear band patterns became statistically indistinguishable from temperature oscillations in the continuum, raising the question of how to evaluate models across the full range of shock strengths.

Target Audience

This paper is best suited for researchers at the intersection of machine learning and computational mechanics — particularly those working on physics-informed machine learning, neural operators, and surrogate modeling for PDE-based simulations. It is also valuable for shock physics and energetic materials researchers seeking fast alternatives to direct numerical simulation, and for machine learning practitioners interested in a candid account of where state-of-the-art models break down on a real scientific problem. The detailed comparison table of model architectures and the physics-driven metrics make it useful for engineers evaluating which AI surrogate approach to adopt.

Authors’ abstract

Modeling shock-to-detonation phenomena in energetic materials (EMs) requires capturing complex physical processes such as strong shocks, rapid changes in microstructural morphology, and nonlinear dynamics of chemical reaction fronts. These processes participate in energy localization at hotspots, which initiate chemical energy release leading to detonation. This study addresses the formation of hotspots in crystalline EMs subjected to weak-to-moderate shock loading, which, despite its critical relevance to the safe storage and handling of EMs, remains underexplored compared to the well-studied strong shock conditions. To overcome the computational challenges associated with direct numerical simulations, we advance the Physics-Aware Recurrent Convolutional Neural Network (PARCv2), which has been shown to be capable of predicting strong shock responses in EMs. We improved the architecture of PARCv2 to rapidly predict shear localizations and plastic heating, which play important roles in the weak-to-moderate shock regime. PARCv2 is benchmarked against two widely used physics-informed models, namely, Fourier neural operator and neural ordinary differential equation; we demonstrate its superior performance in capturing the spatiotemporal dynamics of shear band formation. While all models exhibit certain failure modes, our findings underscore the importance of domain-specific considerations in developing robust AI-accelerated simulation tools for reactive materials.