Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations
2026-07-31 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how to find the exact form of partial differential equations (PDEs) from limited or sparse data. They discover three patterns in how neural networks learn these equations and develop a new method called freeze-then-select that separates the process of fitting the data from choosing the right equation terms. Their approach uses a special adapter to break down the data into spatial and time parts, then applies a stability-based method to pick the correct PDE terms. They test this on different datasets and show it works better than previous methods, especially on complex dynamics like the Kuramoto-Sivashinsky equation.
Partial Differential EquationsNeural NetworksSparse ObservationsEquation DiscoveryOptimization PathCubic SplinesWeak FormStability SelectionGenetic ProgrammingKuramoto-Sivashinsky Equation
Authors
Juncheng Zhong, Chenghuang Shen, Jianfeng Liu, Zhengdong Xiao, Longjiu Luo, Qianrong Wang, Wenjun Xu, Wenlian Lu
Abstract
PDE discovery from sparse observations requires reconstructing a continuous field and selecting the correct differential terms. Our analysis of optimization paths in coupled neural PDE discovery reveals three behaviors: the exact support can persist to the end of training, appear only transiently, or fail to emerge. To decouple equation selection from neural optimization, we develop a freeze-then-select method combining a structured field adapter with Stability-Validated Weak Selection (SVWS). Trained from observations without a PDE residual, the adapter factorizes the field into learned spatial features and temporal coefficients represented by cubic splines. After freezing the field, SVWS identifies recurrent terms across independent weak-form systems, refits candidate supports, and selects the final equation on held-out weak-form systems. Beyond fixed libraries, we apply the same principle to expressions generated by genetic programming and recover the power-law form of an unknown nonlinear diffusion function from sparse, noisy observations. Across all six sparse MDBench regimes, our method attains the highest exact support recovery rate, with its clearest gains over classical and neural baselines on challenging Kuramoto-Sivashinsky dynamics.