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arxiv: 2508.08935 · v4 · submitted 2025-08-12 · 💻 cs.LG

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

Ze Tao , Hanxuan Wang , Fujun Liu This is my paper

Pith reviewed 2026-05-18 23:31 UTC · model grok-4.3

classification 💻 cs.LG
keywords Physics-informed neural networksLiquid residual blocksGating mechanismPredictive accuracyBenchmark problemsPartial differential equationsDeep learningResidual connections
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The pith

Adding liquid residual gating inside hidden layers improves PINN accuracy on benchmarks while keeping training unchanged.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces LNN-PINN, a variant of physics-informed neural networks that inserts a liquid residual gating mechanism only inside the hidden-layer mapping. All other elements of the training process, including how points are sampled, how the loss is formed, and all hyperparameter choices, remain exactly the same as in a standard PINN. On four benchmark problems the modified networks produce lower root-mean-square and mean-absolute errors than the baseline under identical conditions. The authors also report that the same architecture remains stable when the problem dimension, boundary conditions, or the type of differential operator changes.

Core claim

LNN-PINN adds a lightweight liquid residual gating block solely within the hidden-layer mapping of a physics-informed neural network, leaving the sampling strategy, loss composition, and optimization pipeline untouched, and thereby obtains consistent reductions in RMSE and MAE across four benchmark problems together with improved stability across dimensions, boundary conditions, and operator types.

What carries the argument

The liquid residual gating mechanism, a lightweight gating structure placed inside the hidden-layer mapping that modulates residual information flow while preserving the original physics loss and training loop.

If this is right

  • The same gating addition can be inserted into other PINN variants without retraining schedules or loss redesign.
  • Accuracy gains persist when the underlying differential equation changes dimension or boundary type.
  • The method supplies a drop-in architectural fix that requires no extra physics data or modified collocation points.
  • Stability across operator families suggests the gating helps the network represent a wider range of solution manifolds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the gating truly isolates its benefit to internal feature modulation, similar blocks could be tested in other residual-style scientific networks that already use physics losses.
  • The unchanged training pipeline implies the improvement is portable to existing PINN codebases with only a local change to the layer definition.
  • A natural next measurement would be to track how the gating weights evolve during training to see whether they adapt to local solution stiffness.

Load-bearing premise

Any measured accuracy gains come only from the added liquid residual gating and not from any accidental increase in model capacity, change in optimization behavior, or difference in data handling.

What would settle it

Train a standard PINN on the same four benchmarks using identical network depth and width, the exact same random seeds, and the exact same hyperparameter file but without the liquid residual gating; if the resulting RMSE and MAE values match those of LNN-PINN within numerical tolerance, the architectural claim does not hold.

Figures

Figures reproduced from arXiv: 2508.08935 by Fujun Liu, Hanxuan Wang, Ze Tao.

Figure 1
Figure 1. Figure 1: Training workflow of the LNN-PINN framework. Input coordinates 𝐱 are processed through a liquid neural network (LNN) to predict the physical field 𝑢. Automatic differentiation computes the required derivatives, which are combined with boundary/initial conditions to formulate physics-informed residuals. The total loss  = 𝐽PDE + ∑ 𝜆𝑖𝐽𝑖 (where 𝑖 ∈ 𝐷, 𝑁, 𝐼𝐶) is minimized using the Adam optimizer. The iterativ… view at source ↗
Figure 2
Figure 2. Figure 2: Model predictions against ground truth solution for the 1D advection–reaction problem. (a) LNN–PINN prediction (RMSE = 0.001758, MAE = 0.001653) demonstrates significantly closer agreement with the analytical solution than (b) the standard PINN prediction (RMSE = 0.007496, MAE = 0.007442), highlighting the accuracy improvement achieved by the liquid residual gating architecture [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 3
Figure 3. Figure 3: Training loss histories for the 1D advection–reaction problem. (a) LNN–PINN achieves accelerated convergence and superior final loss values, demonstrating the effectiveness of the liquid residual gating architecture. (b) The standard PINN baseline shows comparatively slower convergence and persistently higher residual losses throughout the training process. Z. Tao et al.: Preprint submitted to Elsevier Pag… view at source ↗
Figure 4
Figure 4. Figure 4: Numerical solutions for the 2D Laplace equation with mixed boundary conditions. (a) LNN-PINN prediction demonstrates excellent agreement with the ground truth solution, achieving RMSE = 0.000342 and MAE = 0.000323. (b) Standard PINN prediction shows significantly larger deviations from the reference solution, with RMSE = 0.013116 and MAE = 0.013085. The superior performance of LNN-PINN highlights the effec… view at source ↗
Figure 5
Figure 5. Figure 5: Training loss histories for the 2D Laplace equation with mixed Dirichlet–Neumann boundary conditions. (a) LNN–PINN demonstrates faster convergence and lower final loss values. (b) Standard PINN exhibits slower convergence and higher asymptotic loss under identical training conditions. Z. Tao et al.: Preprint submitted to Elsevier Page 6 of 21 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Predicted and reference temperature fields for non-dimensional steady-state heat conduction in a circular silicon plate with convective boundary conditions. (a) LNN-PINN prediction demonstrates excellent agreement with the reference solution, achieving RMSE = 0.000225 and MAE = 0.000222. (b) Standard PINN prediction shows slightly reduced accuracy with RMSE = 0.000319 and MAE = 0.000315. The superior perfo… view at source ↗
Figure 7
Figure 7. Figure 7: Training loss histories for the non-dimensional steady-state heat conduction problem in a circular silicon plate with convective boundary conditions. (a) LNN-PINN demonstrates accelerated convergence and lower final loss values. (b) Standard PINN exhibits slower convergence and higher asymptotic loss values. interior and the boundary as Ω = {(𝑥𝑖 , 𝑦𝑖 )}𝑁Ω 𝑖=1 and BC = {(𝑥 (𝑏) 𝑗 , 𝑦 (𝑏) 𝑗 )}𝑁BC 𝑗=1 , wher… view at source ↗
Figure 8
Figure 8. Figure 8: Solution accuracy for the anisotropic Poisson–beam equation. (a) LNN–PINN prediction versus ground truth, achieving RMSE = 0.001886 and MAE = 0.001808. (b) Standard PINN prediction versus ground truth, yielding RMSE = 0.007708 and MAE = 0.007658. The significantly lower error metrics demonstrate the superior performance of the proposed LNN–PINN architecture. • L: Dirichlet boundary at 𝑥 = 0, • R: Dirichlet… view at source ↗
Figure 9
Figure 9. Figure 9: Training loss histories for the anisotropic Poisson–beam equation. (a) LNN–PINN demonstrates accelerated convergence and lower final loss values. (b) Standard PINN exhibits slower convergence and higher asymptotic error. Both LNN-PINN and standard PINN are trained for 𝑁train = 5000 iterations using the Adam optimizer with mini-batch sampling. The analytical solution 𝑢 ∗ (𝑥, 𝑦) = 𝑥 2 𝑒 −𝑦 is used exclusivel… view at source ↗
Figure 10
Figure 10. Figure 10: Self-consistent convergence assessment without an analytic reference. On a mesh hierarchy refined with ratio 𝑟 = 2, the coarse solution is projected onto the fine mesh 𝑃ℎ𝑐→ℎ𝑓 𝑇ℎ𝑐 , and the inter-level difference 𝑤 = 𝑇ℎ𝑓 − 𝑃ℎ𝑐→ℎ𝑓 𝑇ℎ𝑐 is exactly integrated on the fine mesh using the P1 mass-matrix formula and constant per-element gradients. The horizontal axis is ℎ ≈ 𝐻max (log scale) and the vertical axis s… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes LNN-PINN, a physics-informed neural network that augments standard PINNs with a liquid residual gating mechanism placed inside the hidden-layer mappings. The central claim is that this lightweight architectural change improves predictive accuracy on PDE problems while leaving the sampling strategy, loss composition, hyperparameter settings, and overall physics-only training pipeline unchanged, with empirical support consisting of lower RMSE and MAE values plus absolute error plots on four benchmark problems.

Significance. If the reported accuracy gains can be shown to arise specifically from the liquid residual gating rather than incidental changes in model capacity or gradient dynamics, the approach would constitute a concise, drop-in architectural refinement that preserves the original PINN optimization and could be readily adopted for improving accuracy in scientific and engineering applications without retraining pipelines.

major comments (2)
  1. [§4 (Experimental evaluation)] §4 (Experimental evaluation): The manuscript does not report the total number of trainable parameters (or FLOPs) for the baseline PINN versus LNN-PINN. Because the liquid residual gating blocks necessarily introduce additional trainable parameters (typically small linear projections or scaling vectors per layer), the observed RMSE/MAE reductions could be explained by a modest increase in effective capacity or altered optimization behavior rather than the specific gating design. An explicit parameter-count comparison or capacity-matched ablation is required to support the abstract's assertion that improvements arise purely from architectural refinement under identical conditions.
  2. [§4, benchmark results] §4, benchmark results: The reported RMSE and MAE decreases across the four problems are presented without error bars, standard deviations from multiple independent runs, or statistical significance tests. This absence makes it difficult to determine whether the accuracy gains are robust or could be attributable to training stochasticity, directly affecting confidence in the central empirical claim.
minor comments (1)
  1. [§3 (Method)] The description of the liquid residual gating mechanism would benefit from an explicit equation or diagram showing how the gating is inserted into the hidden-layer mapping (e.g., the precise form of the residual update and any new activation or scaling terms).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the empirical claims in our work. We address each major point below and revise the manuscript to provide the requested comparisons and statistical reporting.

read point-by-point responses

  1. Referee: [§4 (Experimental evaluation)] §4 (Experimental evaluation): The manuscript does not report the total number of trainable parameters (or FLOPs) for the baseline PINN versus LNN-PINN. Because the liquid residual gating blocks necessarily introduce additional trainable parameters (typically small linear projections or scaling vectors per layer), the observed RMSE/MAE reductions could be explained by a modest increase in effective capacity or altered optimization behavior rather than the specific gating design. An explicit parameter-count comparison or capacity-matched ablation is required to support the abstract's assertion that improvements arise purely from architectural refinement under identical conditions.

    Authors: We agree that parameter counts must be reported to isolate the contribution of the liquid residual gating from any incidental capacity increase. In the revised manuscript we add a table that lists the exact number of trainable parameters for the baseline PINN and LNN-PINN on each of the four benchmarks. The gating blocks introduce only lightweight scaling vectors and small linear projections, producing a 2–4 % increase in total parameters. We further include a capacity-matched ablation in which the hidden-layer widths of the baseline PINN are enlarged to equalize parameter counts; the RMSE/MAE advantages of LNN-PINN persist under these matched conditions, supporting that the gains arise from the gating mechanism itself rather than from extra capacity. revision: yes

  2. Referee: [§4, benchmark results] §4, benchmark results: The reported RMSE and MAE decreases across the four problems are presented without error bars, standard deviations from multiple independent runs, or statistical significance tests. This absence makes it difficult to determine whether the accuracy gains are robust or could be attributable to training stochasticity, directly affecting confidence in the central empirical claim.

    Authors: We acknowledge that variability across runs should be quantified. The revised manuscript now reports mean RMSE and MAE values together with standard deviations obtained from five independent training runs (different random seeds) for every method and benchmark. We also add the results of paired t-tests between baseline PINN and LNN-PINN, with the corresponding p-values included in the updated result tables to demonstrate statistical significance of the observed improvements. revision: yes

Circularity Check

0 steps flagged

No significant circularity in architectural proposal or empirical claims

full rationale

The paper introduces LNN-PINN as an architectural enhancement to PINNs via liquid residual gating and validates it empirically across benchmarks while asserting that sampling, loss, and hyperparameters are held fixed. No derivation chain exists that reduces a claimed prediction or first-principles result to its own inputs by construction. The central assertion that accuracy gains stem from the gating mechanism is framed as an empirical outcome under controlled conditions rather than a self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. The comparison is presented as a direct test of the added blocks, with no evidence of tautological reduction in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, mathematical axioms, or newly postulated physical entities are described in the provided text.

pith-pipeline@v0.9.0 · 5703 in / 1100 out tokens · 35787 ms · 2026-05-18T23:31:27.373019+00:00 · methodology

discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. From Simple to Complex: Curriculum-Guided Physics-Informed Neural Networks via Gaussian Mixture Models

    cs.LG 2026-05 conditional novelty 7.0

    CGMPINN combines Gaussian mixture modeling with curriculum learning to reduce training errors in physics-informed neural networks by up to 97.8% on benchmark PDEs while providing theoretical convergence guarantees.

  2. LSTM-PINN for Steady-State Electrothermal Transport: Preserving Multi-Field Consis tency in Strongly Coupled Heat and Fluid Flow

    physics.comp-ph 2026-04 unverdicted novelty 6.0

    LSTM-PINN uses memory mechanisms to preserve consistency across heat, fluid, and electric fields in electrothermal transport, outperforming standard PINNs on complex convective regimes.

  3. High-Fidelity Reconstruction of Charge Boundary Layers and Sharp Interfaces in Electro-Thermal-Convective Flows via Residual-Attention PINNs

    physics.flu-dyn 2026-04 unverdicted novelty 5.0

    RA-PINN embeds gated attention in a residual network to reduce localized errors at steep charge boundaries while obeying the governing equations.

Reference graph

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