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LNN-PINN: A Unified Physics-Only Training Framework with Liquid Residual Blocks

Ze Tao, Hanxuan Wang, Fujun Liu

Abstract

Physics-informed neural networks (PINNs) have attracted considerable attention for their ability to integrate partial differential equation priors into deep learning frameworks; however, they often exhibit limited predictive accuracy when applied to complex problems. To address this issue, we propose LNN-PINN, a physics-informed neural network framework that incorporates a liquid residual gating architecture while preserving the original physics modeling and optimization pipeline to improve predictive accuracy. The method introduces a lightweight gating mechanism solely within the hidden-layer mapping, keeping the sampling strategy, loss composition, and hyperparameter settings unchanged to ensure that improvements arise purely from architectural refinement. Across four benchmark problems, LNN-PINN consistently reduced RMSE and MAE under identical training conditions, with absolute error plots further confirming its accuracy gains. Moreover, the framework demonstrates strong adaptability and stability across varying dimensions, boundary conditions, and operator characteristics. In summary, LNN-PINN offers a concise and effective architectural enhancement for improving the predictive accuracy of physics-informed neural networks in complex scientific and engineering problems.

LNN-PINN: A Unified Physics-Only Training Framework with Liquid Residual Blocks

Abstract

Physics-informed neural networks (PINNs) have attracted considerable attention for their ability to integrate partial differential equation priors into deep learning frameworks; however, they often exhibit limited predictive accuracy when applied to complex problems. To address this issue, we propose LNN-PINN, a physics-informed neural network framework that incorporates a liquid residual gating architecture while preserving the original physics modeling and optimization pipeline to improve predictive accuracy. The method introduces a lightweight gating mechanism solely within the hidden-layer mapping, keeping the sampling strategy, loss composition, and hyperparameter settings unchanged to ensure that improvements arise purely from architectural refinement. Across four benchmark problems, LNN-PINN consistently reduced RMSE and MAE under identical training conditions, with absolute error plots further confirming its accuracy gains. Moreover, the framework demonstrates strong adaptability and stability across varying dimensions, boundary conditions, and operator characteristics. In summary, LNN-PINN offers a concise and effective architectural enhancement for improving the predictive accuracy of physics-informed neural networks in complex scientific and engineering problems.

Paper Structure

This paper contains 23 sections, 11 theorems, 900 equations, 14 figures, 31 tables.

Table of Contents

  1. Introduction
  2. LNN-PINN: Structural Mechanism and Modeling Framework
  3. Overall Structure of PINNs and Physics-Only Loss
  4. Integrating LNN into PINN: Structural Replacement and Implementation
  5. Numerical Experiments: Method Comparison and Error Assessment
  6. 1D Advection--Reaction (Drift--Decay)
  7. 2D Laplace Equation with Mixed Dirichlet--Neumann Boundary Conditions
  8. Steady-State Heating of a Circular Silicon Plate with Convective Boundary
  9. Anisotropic Poisson--Beam Equation
  10. 2D Steady Relativistic Viscous Fluid Benchmark
  11. 2D Steady Relativistic Viscous Fluid Benchmark with an Alternative Analytical State
  12. Conclusion
  13. Scale Setting and Loss Normalization
  14. Continuous Risk and Monte Carlo Empirical Risk (Measure-Theoretic Setup and Convergence)
  15. Proof (sufficiency for the "Gradient and Gauss-Newton Forms").
  16. ...and 8 more sections

Key Result

Lemma 1

Let $(\Omega,\mathcal{F})$ be a measurable space. Assume $r:\Omega\to\mathbb{R}^m$ is $\mathcal{F}/\mathcal{B}(\mathbb{R}^m)$–measurable and fix $W\in\mathbb{R}^{m\times m}$. Define: With this definition, we obtain an $\mathcal{F}/\mathcal{B}(\mathbb{R})$–measurable function $\phi:\Omega\to\mathbb{R}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (14)

  • Figure 1: Architecture of the LNN--PINN model. The framework preserves the standard physics-only neural solver pipeline while introducing liquid residual blocks as an internal representation mechanism. The network consists of a strictly sequential composition of hidden layers and liquid blocks, where each liquid block performs a width-preserving residual refinement on the latent features generated by the preceding layer. This design isolates architectural enhancement from the governing equations and training formulation, so that the effect of improved hidden-state propagation can be assessed under the same governing equations and training formulation. The resulting model provides a unified and robust mapping from coordinate inputs to physical field predictions across both single-field and multi-field scenarios.
  • Figure 2: Integrated comparison for the 1D advection--reaction problem. The upper panel reports the loss histories, while the lower panel simultaneously presents the analytical field in Eq. \ref{['eq:dd_exact']}, the reconstructed fields, and the corresponding absolute-error distributions for LNN--PINN, RA--PINN, XPINN, and PINN under the same training budget and the same sampling protocol.
  • Figure 3: Integrated comparison for the 2D Laplace equation with mixed Dirichlet--Neumann boundary conditions. The upper panel reports the loss histories, while the lower panel simultaneously presents the analytical field in Eq. \ref{['eq:lap_exact']}, the reconstructed fields, and the corresponding absolute-error distributions for LNN--PINN, RA--PINN, XPINN, and PINN under the same training budget and the same sampling protocol.
  • Figure 4: Integrated comparison for the steady-state heating of a circular silicon plate with convective boundary. The upper panel reports the loss histories, while the lower panel simultaneously presents the reference field from Appendix \ref{['C']}, the reconstructed fields, and the corresponding absolute-error distributions for LNN--PINN, RA--PINN, XPINN, and PINN under the same training budget and the same sampling protocol.
  • Figure 5: Integrated comparison for the anisotropic Poisson--beam equation. The upper panel reports the loss histories, while the lower panel simultaneously presents the analytical field in Eq. \ref{['eq:apbe_exact']}, the reconstructed fields, and the corresponding absolute-error distributions for LNN--PINN, RA--PINN, XPINN, and PINN under the same training budget and the same sampling protocol.
  • ...and 9 more figures

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • Theorem 2
  • ...and 11 more