这篇论文的ESPINN模型用少量快照就能反推出相场模型的化学势和梯度系数，实验稳定，值得关注。
ESPINN框架从瞬态快照数据中反向识别相场模型，同时恢复体化学势和未知梯度系数。在一维Cahn-Hilliard方程实验中，无噪声时单快照对即能准确重建。有噪声时精度下降，但增加快照数量可降低方差、提升鲁棒性。该方法实现了数据高效且物理一致的自由能结构学习。
Extended pseudo-spectral physics-informed neural networks for phase-field models
Phase-field models play a central role in the continuum description of phase separation, in which the bulk free-energy density and the interfacial thickness parameter determine pattern formation and microstructural evolution. In practice, these constitutive quantities are rarely known a priori and must be inferred from limited dynamical observations. In this work, an extended pseudo-spectral physics-informed neural network (ESPINN) framework is developed for the inverse identification of phase-field models from transient snapshot data. It enables the simultaneous recovery of both the bulk chemical potential and unknown gradient coefficients. Numerical experiments on the one-dimensional Cahn-Hilliard equation demonstrate accurate and statistically stable reconstruction in the noiseless regime, with substantial constitutive information recoverable from even a single snapshot pair. In the presence of noise, reconstruction accuracy degrades gracefully, and increasing the number of snapshots improves robustness by reducing variance across runs. These results establish ESPINN as a data-efficient and physically consistent approach for learning free-energy structure in continuum models of phase separation.