How PeRCNN Turns Convolution Kernels into Differential Operators for Physics‑Informed Learning

Why Multiplication Mirrors Physics Better Than Addition

Physical nonlinearities are typically polynomial (e.g., convection term in Navier‑Stokes, cubic coupling in reaction‑diffusion, product of variable and its gradient in Burgers). Standard deep networks generate nonlinearity through additive activation functions, which produce nested functions that do not align with the polynomial structure of PDEs. PeRCNN’s insight is to use multiplication directly, so the learned model is inherently interpretable as a polynomial matching the PDE form.

PeRCNN Design Blueprint

2.1 Π‑block: Multiplication Leads to Polynomials

The Π‑block consists of several parallel convolution layers; each extracts spatial features from the state. Their outputs are fused by element‑wise product (Π) instead of an activation or residual addition, followed by a linear combination via a convolution. The expressive power grows with the number of parallel channels: with two channels the block can represent all second‑order polynomials, with three channels it can cover third‑order terms. Theorem 1 proves that, given enough channels and parallel layers, a Π‑block can approximate any continuous function to arbitrary precision, matching the structure of polynomial‑type PDEs.

2.2 Convolution Filters as Spatial Differencers

In PeRCNN, convolution kernels are not visual feature detectors but finite‑difference templates. For example, a second‑order central difference Laplacian corresponds to a specific kernel. When a PDE term is known (e.g., diffusion), PeRCNN creates a fixed “physics bypass” convolution that implements the exact difference stencil, leaving the Π‑block to learn only the unknown terms. This splits the learning burden, compressing the parameter search to the truly unknown dynamics.

2.3 Physical Padding for IC/BC

Initial conditions (IC) are fed as the first input to the recurrent network; low‑resolution ICs are up‑sampled by an Initial State Generator (ISG). Boundary conditions (BC) are enforced at each time step via physics‑based padding: Dirichlet values are directly written, Neumann values are computed from normal derivatives, and periodic boundaries are filled by wrapping. Consequently, the loss function contains only the PDE residual, and IC/BC are satisfied structurally.

2.4 Recursive Time Advancement

PeRCNN adopts explicit time‑integration schemes. Using forward Euler as a baseline, the Π‑block and physics bypass together compute the next state. The paper also employs Runge‑Kutta for higher accuracy. Lemma 2 shows that the local truncation error of the forward scheme diminishes with the time step, guaranteeing convergence comparable to classic finite‑difference methods.

Numerical Experiments

3.1 Forward Problem Accuracy

Four reaction‑diffusion systems (2D λ‑Ω, 2D/3D FitzHugh‑Nagumo, 2D Gray‑Scott) were solved. Baselines were ConvLSTM and PINN. In 2D the error gap was modest, but in 3D ConvLSTM and PINN errors rose sharply while PeRCNN maintained low RMSE, demonstrating the advantage of full‑field convolution in higher dimensions.

3.2 Inverse Problem Coefficient Identification

On the 2D Gray‑Scott system with six unknown coefficients, two data scenarios were tested. In the multi‑snapshot low‑resolution scenario (S1) the average absolute relative error (MARE) was 0.60 % without noise and 1.61 % with 10 % Gaussian noise. In the sparse‑snapshot scenario (S2) MARE was 0.96 % (no noise) and 1.05 % (10 % noise), outperforming the PINN baseline.

3.3 Data‑Driven Modeling and Extrapolation

PeRCNN was trained on 41 noisy low‑resolution frames of a 2D Gray‑Scott system and required to reconstruct 801 high‑resolution frames and then extrapolate 1 700 steps forward. Competing models (Recurrent ResNet, ConvLSTM, PDE‑Net, Deep Hidden Physics Model) either failed to reconstruct high‑resolution fields or diverged during extrapolation. Only PeRCNN kept a stable, low‑error trajectory in both training and extrapolation phases.

3.4 Equation Discovery

The discovery pipeline consists of (1) data reconstruction with PeRCNN, (2) sparse regression (STRidge) on the reconstructed data to select active terms, and (3) coefficient refinement by re‑encoding the discovered PDE into PeRCNN and fine‑tuning. On the 2D Gray‑Scott system, PeRCNN recovered the exact PDE terms and coefficients under no noise and 5 % noise; even with 10 % noise it identified all major terms, adding only a near‑zero redundant term.

Interpretability via Symbolic Extraction

The Π‑block’s multiplicative structure allows direct extraction of analytic expressions from trained parameters using SymPy. For a 3D Gray‑Scott model, the reaction terms extracted matched the true coefficients, while diffusion terms were handled by the fixed physics bypass. Similar success was reported for a 2D Burgers equation, where the viscous coefficient was recovered within ~2 % error.

Constraints and Frontiers

Current implementation requires data on regular Cartesian grids; irregular geometries would need graph convolutions, which are not yet realized.

The universal approximation theorem assumes the target function is polynomial‑approximable; PDEs involving division, exponentials, or trigonometric functions may need many channels or additional symbolic activation layers.

Scaling to high‑resolution 3D engineering problems will demand temporal batching and multi‑GPU parallelism.

Baseline PINN comparisons used a vanilla configuration; newer PINN variants with adaptive weighting or causal training could narrow the performance gap.

Reviewers and Follow‑Up Work

The paper was reviewed by Ilias Bilionis (Purdue) and Lu Lu (DeepXDE/DeepONet). The senior authors previously introduced PINN‑SR, showing a progression from continuous DNN‑based equation discovery to PeRCNN’s discrete, physics‑encoded approach.

References

Rao C, Ren P, Wang Q, et al. Encoding physics to learn reaction–diffusion processes. Nature Machine Intelligence , 2023, 5:765‑779.

Raissi M, Perdikaris P, Karniadakis GE. Physics‑informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics , 2019, 378:686‑707.

Wang S, Teng Y, Perdikaris P. Understanding and mitigating gradient flow pathologies in physics‑informed neural networks. SIAM Journal on Scientific Computing , 2021, 43(5):A3055‑A3081.

Long Z, Lu Y, Ma X, Dong B. PDE‑Net: Learning PDEs from data. ICML , 2018:3208‑3216.

Ren P, Rao C, Liu Y, et al. PhyCRNet: Physics‑informed convolutional‑recurrent network for solving spatiotemporal PDEs. Computer Methods in Applied Mechanics and Engineering , 2022, 389:114399.

Chen Z, Liu Y, Sun H. Physics‑informed learning of governing equations from scarce data. Nature Communications , 2021, 12:6136.

Brunton SL, Proctor JL, Kutz JN. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences , 2016, 113(15):3932‑3937.

Li Z, Kovachki N, Azizzadenesheli K, et al. Fourier Neural Operator for Parametric Partial Differential Equations. ICLR , 2021.

Lu L, Jin P, Pang G, Zhang Z, Karniadakis GE. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence , 2021, 3:218‑229.

Shi X, Chen Z, Wang H, et al. Convolutional LSTM network: A machine learning approach for precipitation nowcasting. NeurIPS , 2015:802‑810.