Shrinkage-induced surface cracking is common in drying mud, paints, and coatings. The statistical properties of these patterns offer valuable insights into their formation and evolution, and the distribution of fragment sizes is particularly informative. Accurately estimating this distribution from limited data remains a central challenge in inverse modeling and materials design. Among these statistical properties, a dynamical scaling law for the distribution of fragment sizes illustrates how order arises in seemingly random cracking patterns. As fracturing proceeds, the mean size decreases and the distribution evolves. When the sizes are normalized by the mean at each time, the distributions at different times converge to a single time-invariant distribution, which is a hallmark of dynamical scaling. This behavior has been observed in numerical simulations and experiments, which suggests that shrinkage-induced fragmentation may exhibit universal features. These insights enable the retrospective estimation of distribution shapes, support predictions of future evolution, and help elucidate the underlying physical mechanisms.

The stochastic model proposed by Ito and Yukawa (2014) reproduces these dynamical features and leads to an integro-differential equation (IDE), the solution of which is the time-invariant distribution. Although this IDE is typically solved using grid-based methods such as finite differences, its numerical stiffness can introduce grid-dependent artifacts. Consequently, obtaining reliable solutions requires extensive computations and careful validation. These limitations are prohibitive for inverse analyses that require repeated evaluations of the distribution over a range of physical parameter settings. Studies using real fragment data have thus remained limited.

In this study, we introduce a physics-informed neural network approach to address these limitations. The network encodes the governing physics directly and is trained to satisfy the IDE as well as the constraints that a probability density must satisfy. This formulation mitigates spurious numerical artifacts observed in grid-based methods, reduces overtraining, and promotes generalization. After training, the network takes physical parameters as input and directly produces a time-invariant distribution of fragment sizes, which enables rapid prediction without the need to solve the IDE explicitly. In numerical tests, the proposed method was approximately 2,500 times faster than a grid-based baseline technique while maintaining high accuracy.

With this speed and accuracy, the proposed physics-informed network provides a reusable surrogate for near-instantaneous predictions and systematic parameter exploration. By serving as a forward model for Monte Carlo sampling, the framework supports Bayesian inference to quantify uncertainty and enables inverse analyses that reconstruct shrinkage histories and constrain physical parameters. We expect the proposed method to be useful in industrial and geotechnical contexts such as experimental design, sensitivity analysis, model comparison, and tests of universality across materials.

(written by Shin-ichi Ito on behalf of all authors.)

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