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

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural PDE solvers · 0 citations (OpenAlex)

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Proposes Mollifier Layers, an output-layer convolution with analytic mollifier derivatives that replaces recursive autodiff for high-order derivatives in physics-informed inverse PDE learning, and applies it to chromatin reaction-rate inference.

Problem

Physics-informed ML for inverse PDE parameter estimation relies on recursive autodiff, which is memory-hungry, slow, and inaccurate for high-order derivatives, especially with noisy data.

Approach

Instead of autodiff, the base network predicts a generator g, and u and its derivatives are obtained by convolving g with analytic derivatives of a compactly supported smooth mollifier at a single output layer, so derivative computation is decoupled from network depth and acts as a smoothing operation.

Key results

• On the 2D reaction-diffusion (fourth-order) task, mollified PINN reduced training time from 3386 s to 335 s and peak memory from 2.75 GB to 0.23 GB, with Laplacian correlation 0.78 vs 0.21 for PINN.
• On the 2D heat equation, mollified PINN raised Laplacian correlation from 0.16 to 0.99 and mean correlation from 0.81 to 0.99 with lower memory (0.24 GB vs 1.20 GB).
• Mollified PirateNet on the reaction-diffusion task achieved 0.99 mean correlation and 125 s training time versus 817 s for PirateNet.

Surprising findings

• The paper reports that a native PINN's Laplacian correlation is only 0.21 in the reaction-diffusion system, showing autodiff failures at high order.
• Native PINNsFormer shows a temporal correlation of only 0.35 on the Langevin task despite a high mean correlation.

Limitations

• Performance is sensitive to the mollifier kernel choice (order and support), which is hand-tuned.
• The implementation is limited near boundaries and on anisotropic grids.
• PINNsFormer was not paired with a mollified version due to computational cost.

Experimental setup

Datasets: synthetic PDE simulations (Langevin, heat, reaction-diffusion) STORM super-resolution chromatin imaging of human nuclei

Benchmarks: 1D Langevin equation 2D heat equation 2D reaction-diffusion equation STORM chromatin imaging

Metrics: mean correlation temporal correlation spatial correlation Laplacian correlation peak memory training time

Baselines: PINN PirateNet PINNsFormer

Builds on: PINNs PirateNet PINNsFormer Savitzky-Golay filters

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