Extended pseudo-spectral physics-informed neural networks for phase-field models

Technical Summary: Extended Pseudo-Spectral Physics-Informed Neural Networks for Phase-Field Models

Problem Statement
Phase-field models are central to the continuum description of phase separation, where pattern formation and microstructural evolution are governed by the bulk free-energy density and interfacial thickness parameters. In practical applications, such as materials science and cell biology, these constitutive quantities are rarely known a priori and must be inferred from limited dynamical observations. While classical inverse methods and recent Physics-Informed Neural Network (PINN) approaches have been applied to phase-field models, existing techniques often face limitations. Specifically, previous work by Zhao [31] utilized pseudo-spectral PINNs (SPINNs) to identify the bulk chemical potential but required the interfacial thickness parameter and mobility to be fixed a priori. Furthermore, treating all constitutive quantities as unknowns in classical optimization often leads to inherent non-identifiability due to scaling degeneracy. The core challenge addressed in this work is the simultaneous inverse identification of both the bulk chemical potential (the derivative of the homogeneous free energy density) and unknown gradient coefficients (specifically the interfacial thickness parameter) from transient snapshot data, without relying on equilibrium profiles alone.

Methodology: Extended SPINN (ESPINN)
The authors propose the Extended Pseudo-Spectral Physics-Informed Neural Network (ESPINN) framework. This method extends the SPINN architecture by treating physical parameters as trainable variables alongside the neural network approximating the free energy function.

1. Mathematical Formulation: The framework targets general phase-field PDEs of the form $\partial_t \phi = G [\sum \mu_j g_j(\phi, \nabla \phi, \Delta \phi) + f(\phi)]$, where $f(\phi)$ (bulk chemical potential) and coefficients $\mu_j$ (gradient terms) are unknown. The authors focus on the Cahn-Hilliard and Allen-Cahn equations, where the unknown gradient coefficient corresponds to the interfacial thickness parameter $\epsilon$.
2. Neural Architecture:
   • Bulk Chemical Potential: The function $f(\phi)$ is approximated by a Multi-Layer Perceptron (MLP).
   • Scalar Parameters: Unknown coefficients (e.g., $\epsilon^2$) are parameterized by single trainable scalars $\theta$ using a sigmoid reparameterization to enforce physical bounds ($\gamma_{lb} \leq \gamma \leq \gamma_{ub}$).
   • Loss Function: The training objective minimizes the discrepancy between observed data pairs $(\Phi^{(1)}, \Phi^{(2)})$ separated by time steps $\Delta t$ and the predictions of a semi-implicit linear stabilized scheme. The loss function includes a hyperparameter $C_0$ to condition the optimization, adjusted based on data noise levels.
3. Optimization Strategy:
   • Noiseless Data: The framework employs a two-stage optimization: Adam followed by L-BFGS to exploit the smoothness of the objective landscape for precise convergence.
   • Noisy Data: To prevent overfitting to stochastic perturbations, the L-BFGS stage is omitted, and optimization relies solely on Adam with a reduced loss multiplier ($C_0$) to moderate gradient magnitudes.

Key Contributions

• Simultaneous Inference: The primary contribution is the extension of the SPINN framework to simultaneously recover the bulk chemical potential and unknown gradient coefficients (interfacial thickness) from transient data, overcoming the need to fix these parameters a priori.
• Data Efficiency: The framework demonstrates that substantial constitutive information is encoded in even a single snapshot pair, allowing for accurate reconstruction with minimal temporal data.
• Robustness to Noise: The study systematically investigates the framework's performance under noise, demonstrating that increasing the number of snapshot pairs acts as an implicit regularizer, reducing variance and improving accuracy without altering the deterministic features of the learned model.
• Architectural Sensitivity Analysis: The paper provides a detailed analysis of activation functions and network depth, revealing that the smoothness and differentiability of the activation function are more critical to reconstruction accuracy than network capacity (depth or width).

Results
Numerical experiments were conducted on one-dimensional Cahn-Hilliard simulations using both polynomial and logarithmic free energy densities.

• Noiseless Regime: The ESPINN framework achieved accurate and statistically stable reconstruction of both the bulk chemical potential and the interfacial thickness parameter. Remarkably, using only a single snapshot pair ($N=1$) yielded results closely matching the true functions. Increasing the number of snapshots ($N$) to 2 eliminated visible variance in the reconstruction. Further increases in $N$ up to 64 improved statistical stability, though diminishing returns were observed beyond $N=64$.
• Noisy Regime: In the presence of multiplicative noise, reconstruction accuracy degraded gracefully. While limited noisy data ($N=1$) resulted in high variability, increasing $N$ systematically reduced errors and variance. The method remained stable even at higher noise levels ($\delta_L = 10^{-2}$), though a visible bias emerged.
• Architectural Findings: Smooth activation functions (Sigmoid, Tanh, SiLU, GELU) consistently outperformed ReLU, which yielded errors up to four orders of magnitude higher. Network depth and width had minimal impact on final accuracy, suggesting the inverse problem is not capacity-limited but rather dependent on the differentiability properties of the activation function.
• Learning Rate Sensitivity: The method showed sensitivity to learning rates in the Adam optimizer. An exponentially decaying learning rate (initially $10^{-2}$, decaying to $10^{-4}$) provided the best balance of convergence speed and stability.

Significance and Claims
The authors claim that ESPINN establishes a data-efficient and physically consistent approach for learning free-energy structures in continuum models of phase separation. The framework successfully addresses the inverse problem of identifying constitutive quantities from transient dynamics, a task where equilibrium profiles alone are insufficient due to scaling degeneracy. By demonstrating that accurate recovery is possible from limited data and that the method remains robust in noisy conditions, the work suggests that transient dynamical data provide a reliable basis for inferring constitutive information. The authors note that while the current study is restricted to one-dimensional models with constant mobility, the methodology naturally extends to higher dimensions and more complex free-energy functionals, offering a practical tool for coarse-grained modeling in biomolecular condensates, polymer blends, and alloy systems.