Penn Stabilizes Inverse PDEs

Penn researchers proposed Mollifier Layers to reduce the bottleneck of repeated automatic differentiation in inverse PDE inference under heavy noise. The paper was published in Transactions on Machine Learning Research, with a presentation scheduled at NeurIPS 2026.

The core idea is that instead of repeatedly differentiating directly on the neural network output, the signal is first smoothed with a mollifier, then derivatives are computed via fixed convolution-based operations. This layer is lightweight and can be attached to various physics-informed models, improving stability against high-order derivatives and noise.

In benchmarks, it consistently outperformed standard physics-informed neural networks (PINNs).

• 1D Langevin: temporal correlation 0.97 vs 0.36, memory 0.16GB vs 0.21GB, time 1,615s vs 2,138s
• 2D heat equation: spatial correlation 0.99 vs 0.21, memory 0.24GB vs 1.20GB
• 4th-order reaction-diffusion: time 335s vs 3,386s, memory 0.23GB vs 2.75GB, average parameter correlation 0.99 vs 0.44

The researchers explained that they reduced memory and training time by 6 to 10 times. Applications include chromatin inverse inference, heat transfer, fluid dynamics, and noisy dynamical systems, while limitations remain in kernel selection, boundary conditions, and anisotropic grids.