Deep Wavelet Diffeomorphism: A New Approach to Manifold Regularization for PINN Training

Physics-Informed Neural Networks (PINNs) are a mesh-free, data-driven approach for solving partial differential equations (PDEs). However, their accuracy in capturing strong-gradient regions (such as shock waves) depends on dense, uniform collocation points, which can lead to convergence issues and susceptibility to local optima. Traditional adaptive wavelet solvers excel at identifying discontinuous regions but suffer from time-step constraints and require cumbersome threshold tuning. To address these challenges, this paper proposes a Deep Wavelet Diffeomorphism (DWD) framework that integrates the strengths of both methods through a decoupled mechanism involving manifold regularization and low-dimensional training. Leveraging interpolating multiresolution analysis, DWD constructs a deep wavelet diffeomorphism that maps distorted physical-space training manifolds into smooth computational-space manifolds, while generating an optimal non-uniform collocation set---dense in feature-rich regions and sparse in smooth regions. This manifold mapping discretizes high-dimensional PDEs into a low-dimensional collocation system, which is then input into a standard PINN for efficient and stable optimization. Experimental results demonstrate that DWD achieves accuracy comparable to uniform super-resolution PINNs, while reducing collocation points by an order of magnitude and shortening training time by 5 to 8 times. It outperforms pure wavelet solvers and traditional PINNs in both accuracy and robustness. Concise and compatible with existing deep learning pipelines, DWD offers a data-efficient, low-dimensional discretization approach for high-dimensional PDEs and inverse problems in physics-informed learning.