Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning

TL;DR

Mollifier Layers provide an efficient, noise-robust method for computing high-order derivatives in physics-informed neural networks, improving accuracy and scalability in inverse PDE problems.

Contribution

We introduce Mollifier Layers, a novel convolutional module that replaces autodiff for derivative computation, enhancing efficiency and robustness in physics-informed learning.

Findings

Significant memory and time savings in training

Improved accuracy in high-order PDE parameter estimation

Successful application to biomedical inverse problems

Abstract

Parameter estimation in inverse problems involving partial differential equations (PDEs) underpins modeling across scientific disciplines, especially when parameters vary in space or time. Physics-informed Machine Learning (PhiML) integrates PDE constraints into deep learning, but prevailing approaches depend on recursive automatic differentiation (autodiff), which produces inaccurate high-order derivatives, inflates memory usage, and underperforms in noisy settings. We propose Mollifier Layers, a lightweight, architecture-agnostic module that replaces autodiff with convolutional operations using analytically defined mollifiers. This reframing of derivative computation as smoothing integration enables efficient, noise-robust estimation of high-order derivatives directly from network outputs. Mollifier Layers attach at the output layer and require no architectural modifications. We compare them with three distinct architectures and benchmark performance across first-, second-, and fourth-order PDEs -- including Langevin dynamics, heat diffusion, and reaction-diffusion systems -- observing significant improvements in memory efficiency, training time and accuracy for parameter recovery across tasks. To demonstrate practical relevance, we apply Mollifier Layers to infer spatially varying epigenetic reaction rates from super-resolution chromatin imaging data -- a real-world inverse problem with biomedical significance. Our results establish Mollifier Layers as an efficient and scalable tool for physics-constrained learning.