Towards Robust Surrogate Models: Benchmarking Machine Learning Approaches to Expediting Phase Field Simulations of Brittle Fracture

TL;DR

This work tackles the challenge of building robust ML surrogates for phase-field fracture modeling by introducing the PFM-Fracture benchmark, a large, diverse dataset of 6,000 two-dimensional quasi-static simulations that span three energy decomposition schemes and two boundary conditions. It evaluates three representative baselines—PINN, FNO, and UNet—and several ensembling strategies, revealing that PINNs struggle with complex fracture paths while UNet and FNO offer more reliable predictions, with stacking ensembles providing the largest gains for FNO. The dataset and open-source code provide a standardized testbed to stress-test and compare ML approaches in fracture mechanics and related solid-mechanics problems, facilitating systematic improvements and fair benchmarking. The study highlights the limitations of common metrics like MSE for fracture prediction and advocates for topology-aware evaluation, while signaling clear directions for future work in data-efficient physics-informed models and advanced ensembling techniques. Overall, the work contributes a valuable benchmark and baseline results that can accelerate progress in data-driven fracture modeling and multi-scale uncertainty quantification in engineering systems.

Abstract

Data driven approaches have the potential to make modeling complex, nonlinear physical phenomena significantly more computationally tractable. For example, computational modeling of fracture is a core challenge where machine learning techniques have the potential to provide a much needed speedup that would enable progress in areas such as mutli-scale modeling and uncertainty quantification. Currently, phase field modeling (PFM) of fracture is one such approach that offers a convenient variational formulation to model crack nucleation, branching and propagation. To date, machine learning techniques have shown promise in approximating PFM simulations. However, most studies rely on overly simple benchmarks that do not reflect the true complexity of the fracture processes where PFM excels as a method. To address this gap, we introduce a challenging dataset based on PFM simulations designed to benchmark and advance ML methods for fracture modeling. This dataset includes three energy decomposition methods, two boundary conditions, and 1,000 random initial crack configurations for a total of 6,000 simulations. Each sample contains 100 time steps capturing the temporal evolution of the crack field. Alongside this dataset, we also implement and evaluate Physics Informed Neural Networks (PINN), Fourier Neural Operators (FNO) and UNet models as baselines, and explore the impact of ensembling strategies on prediction accuracy. With this combination of our dataset and baseline models drawn from the literature we aim to provide a standardized and challenging benchmark for evaluating machine learning approaches to solid mechanics. Our results highlight both the promise and limitations of popular current models, and demonstrate the utility of this dataset as a testbed for advancing machine learning in fracture mechanics research.

Figures (17)

• Figure 1: Comparison between approaches to evaluating surrogate model efficacy: a) Standard examples from the literature such as tension, shear, and crack coalescence goswami2020adaptivemanav2024phase, b) Our proposed benchmark dataset designed to provide a diverse set of challenges across initial conditions, load conditions, and energy decomposition methods.
• Figure 2: A general schematic of a mechanical system with a) the real sharp crack surface $\Gamma$ inside a domain $\Omega$ with boundary conditions, and b) the crack approximation via the scalar phase field with $\phi = 1$ in the cracked region and $\phi=0$ in the intact region with a diffuse transition between the regions.
• Figure 3: Geometrical description and boundary conditions of the three benchmark cases and the resulting crack path and force displacement curves to validate the implemented phase field model. Here we show: a) the Tension case, b) the Shear case and c) the crack coalescence case under uniaxial tension. The crack patterns for the spectral decomposition is depicted in (ii). The force displacement curve for all three of our energy decomposition approaches are reported in (iii) alongside results from the literature.
• Figure 4: Visualization of crack branching near the boundaries for volumetric-deviatoric and star-convex decomposition. Note the qualitative differences between the three approaches.
• Figure 5: Overview of our dataset: a) Schematic showing the boundary conditions and the geometric properties of the domain, with $n$ number of cracks, where the $\theta$ orientation and center coordinate $(x, y)$ of each crack is randomly sampled for each simulation case. The crack length $l = 0.25$ mm is fixed in all simulations, b) For each simulation, phase field and x and y components of the displacement in multiple loading steps and the boundary force vs. boundary displacement curves are saved.