cs.LGSep 19, 2026

LPINNs: First-Layer Gated Localization for Physics-Informed Neural Networks

Authors: Lakshay Chawla, Hardik Jain

Organizations: Indian Institute of Technology Jodhpur

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Abstract

Physics-informed neural networks (PINNs) use one shared representation over the computational domain, which can become difficult to optimize on long domains and for high-order operators. We study a minimal alternative: multiply the first hidden activation of an otherwise unchanged dense PINN by input-dependent localization functions, giving first-layer units receptive fields without partitioning the domain or adding interface losses. We screen 13 families of localization functions, in up to three parameterizations each, on a nonlinear harmonic oscillator (HO), a heat equation on a long spatial interval, and a manufactured four-dimensional (4D) fourth-order problem, with ten paired seeds throughout. Three configurations give large reductions in solution error at matched budgets: (i) Fixed Gaussian localization functions on the 2π2π HO domain cut mean solution RMSE from 4.8369×10−14.8369\times10^{-1} to 8.83×10−38.83\times10^{-3} at 3k epochs. (ii) The inverse-quadratic family with learnable centers and widths cuts it from 2.896×10−12.896\times10^{-1} to 3.06×10−23.06\times10^{-2} on the 8π8π heat domain at 10k epochs. (iii) Fixed bump localization functions cut it from 1.75947×1011.75947\times10^{1} to 2.260×10−12.260\times10^{-1} on the 4π4π 4D domain at 10k epochs. Every paired seed improves in these three comparisons. The screen also shows that the mechanism is not a free win: on HO only 2 of 13 families beat the baseline, and 10 of the remaining 11 are 9 to 23 times worse; on 4D four families are non-finite and five are more than three orders of magnitude worse than the baseline. The inverse-quadratic family is the only one that beats the baseline on all three equations. Overall, these results show that first-layer localization can provide measurable improvements to baseline PINNs on long-domain and high-order problems.

Figures & tables

ProblemDomain valuesEpoch budgets
HOt∈[0,L], L∈{π,2π,3π,4π}t\in[0,L],\ L\in\{\pi,2\pi,3\pi,4\pi\}11 k, 33 k, 55 k, 1010 k, 5050 k, 100100 k
Heatx∈[0,L], t∈[0,1]x\in[0,L],\ t\in[0,1] L∈{π,2π,4π,8π}L\in\{\pi,2\pi,4\pi,8\pi\}11 k, 33 k, 55 k, 1010 k, 5050 k
4Dx∈[0,L]4, L∈{π,2π,4π}x\in[0,L]^{4},\ L\in\{\pi,2\pi,4\pi\}11 k, 33 k, 55 k, 1010 k
Table 1: Domain sizes and epoch budgets used in the localization campaign.
Localization FunctionEpochsFinal Residual LossResidual RMSESolution RMSESolution MAE
Baseline3k5.010×10−25.010\times 10^{-2}1.1995×10−11.1995\times 10^{-1}4.8369×10−14.8369\times 10^{-1}4.0788×10−14.0788\times 10^{-1}
Gaussian3k1.85×10−31.85\times 10^{-3}2.639×10−22.639\times 10^{-2}8.83×10−38.83\times 10^{-3}7.06×10−37.06\times 10^{-3}
Baseline1k6.230×10−26.230\times 10^{-2}1.7438×10−11.7438\times 10^{-1}6.7583×10−16.7583\times 10^{-1}5.5689×10−15.5689\times 10^{-1}
Gaussian1k1.1420×10−11.1420\times 10^{-1}3.0155×10−13.0155\times 10^{-1}2.14721×1002.14721\times 10^{0}1.79705×1001.79705\times 10^{0}
Table 2: HO at L=2πL=2\pi with 64×6464\times 64 networks and fixed Gaussian localization functions. Means over ten paired seeds. All metric entries are shown in scientific notation. Bold marks the localized solution RMSE and MAE at the larger budget.
Figure 1: HO at L=2πL=2\pi and 3k epochs: recovered trajectories with the numerical reference (left), residual-loss histories (center), and absolute residuals (right). Thin curves are the ten paired seeds and thick curves are their means, with loss averaged in log⁡10\log_{10} space.
Localization FunctionEpochsFinal Residual LossResidual RMSESolution RMSESolution MAE
Baseline3k6.433×10−16.433\times 10^{-1}7.073×10−17.073\times 10^{-1}2.909×10−12.909\times 10^{-1}2.328×10−12.328\times 10^{-1}
Inverse-quadratic3k1.208×10−11.208\times 10^{-1}3.353×10−13.353\times 10^{-1}1.011×10−11.011\times 10^{-1}7.17×10−27.17\times 10^{-2}
Baseline10k5.212×10−15.212\times 10^{-1}7.047×10−17.047\times 10^{-1}2.896×10−12.896\times 10^{-1}2.319×10−12.319\times 10^{-1}
Inverse-quadratic10k2.51×10−22.51\times 10^{-2}1.499×10−11.499\times 10^{-1}3.06×10−23.06\times 10^{-2}2.24×10−22.24\times 10^{-2}
Table 3: Heat at L=8πL=8\pi with 64×6464\times 64 networks and inverse-quadratic localization functions with learnable centers and widths. Means over ten paired seeds. All metric entries are shown in scientific notation. Bold marks the localized solution RMSE and MAE at the larger budget.
Figure 2: Heat equation at L=8πL=8\pi and 10k epochs for both models: solution at t=0.5t=0.5 (left), residual-loss histories (center), and absolute residuals at t=0.5t=0.5 (right). Thin curves are the ten paired seeds and thick curves are their means, with loss averaged in log⁡10\log_{10} space.
Localization FunctionEpochsFinal Residual LossResidual RMSESolution RMSESolution MAE
Baseline3k2.6200×10−32.6200\times 10^{-3}5.3494×10−25.3494\times 10^{-2}1.420850×1021.420850\times 10^{2}1.111457×1021.111457\times 10^{2}
Bump3k2.5200×10−32.5200\times 10^{-3}1.114×10−31.114\times 10^{-3}9.221×10−19.221\times 10^{-1}5.287×10−15.287\times 10^{-1}
Baseline10k5.34×10−55.34\times 10^{-5}7.320×10−37.320\times 10^{-3}1.75947×1011.75947\times 10^{1}1.40520×1011.40520\times 10^{1}
Bump10k4.72×10−54.72\times 10^{-5}3.44×10−43.44\times 10^{-4}2.260×10−12.260\times 10^{-1}1.296×10−11.296\times 10^{-1}
Table 4: 4D at L=4πL=4\pi with 32×3232\times 32 networks and fixed bump localization functions. Means over ten paired seeds. All metric entries are shown in scientific notation; the exponents are given inside the cells. Bold marks the localized solution RMSE and MAE at the larger budget.
Figure 3: 4D problem at L=4πL=4\pi and 10k epochs for both models: normalized solution on the diagonal slice (left), empirical distribution of relative error (center), and residual-loss histories (right). Thin curves are the ten paired seeds and thick curves are their means, with loss averaged in log⁡10\log_{10} space.
Localization FunctionHOHeat4D
Baseline9.946×10−19.946\times 10^{-1}8.386×10−28.386\times 10^{-2}3.126×1023.126\times 10^{2}
Gaussian8.402×10−58.402\times 10^{-5} (F)7.956×10−27.956\times 10^{-2} (F)n/a
Activated1.788×1011.788\times 10^{1} (S)7.502×10−27.502\times 10^{-2} (S)1.777×1071.777\times 10^{7} (F)
Super-Gaussian1.541×1011.541\times 10^{1} (S)7.470×10−27.470\times 10^{-2} (M)1.621×1071.621\times 10^{7} (S)
Ricker1.148×1011.148\times 10^{1} (F)8.339×10−28.339\times 10^{-2} (F)n/a
Boxcar9.1409.140 (S)7.237×10−27.237\times 10^{-2} (S)1.346×1071.346\times 10^{7} (S)
Table 5: Screen of localization-function families at the primary settings: HO at 2π2\pi /3k, Heat at 8π8\pi /10k, and 4D at 4π4\pi /10k. Entries are mean solution MSE over ten seeds; lower is better. F, S, and M denote fixed, learnable-width, and learnable-center/width parameterizations; each cell gives the lowest finite parameterization for that family, with the winning variant in parentheses. n/a means that no tested parameterization produced a finite solution metric for that problem. Bold marks the lowest valid entry in each problem column. Columns have different solution scales and should be compared only within a column.
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