cs.LGAug 24, 2026

When Does Frequency Decomposition Benefit Physics-Informed Neural Networks? A Preliminary Ablation Study

Authors: Shubham Rai

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Abstract

Partial differential equations (PDEs) often have high-frequency and multi-scale features that neural networks struggle to approximate. Physics-Informed Neural Networks (PINNs) build the governing equations directly into training, but suffer from spectral bias: they learn low-frequency components faster than high-frequency ones. Techniques such as Fourier feature embeddings and sinusoidal activations address this, but most studies assume they help across the board without checking which spectral regimes actually benefit. We introduce a dual-branch, spectrally-gated architecture (DBSG-PINN) that splits low- and high-frequency components into separate subnetworks joined by an adaptive gate, and use it to run a partially controlled ablation of frequency decomposition and spectral routing. We test this on five one-dimensional benchmark PDEs, ranging from smooth, single-scale problems to oscillatory, multi-scale ones. Frequency decomposition helps most on the spectrally complex benchmarks, cutting relative L2L_2 error by up to 59.2%59.2\% on a multimodal wave problem, but gives little benefit on smoother PDEs. On one benchmark (1D Wave), it performs substantially worse than a simpler fixed-combination variant. The gate's benefit scales with how spectrally rich the target solution is: the full model's advantage over the ablations is largest on multi-scale benchmarks and smallest (or negative) on single-scale ones, consistent with the gate exploiting frequency structure rather than acting as noise,though we do not directly visualize or quantify its spatial activations in this study. All results come from a single training seed across five 1D benchmarks, so we present this as an exploratory study meant to raise questions rather than answer them, and outline the additional seeds and benchmarks needed to test whether the pattern holds.

Figures & tables

Figure 1: DBSG-PINN architecture
ModelRelative L2 Error ↓\downarrowL∞L_{\infty} Error ↓\downarrowMean PDE Residual ↓\downarrowSpectral Error ↓\downarrowHF Recovery ↑\uparrowLF Recovery ↑\uparrow
DBSG-PINN (Full)0.07620.16320.02180.03710.86030.9658
NoGate0.11610.22350.02060.08400.80190.9239
LowOnly0.18690.46490.03780.13870.72350.8798
HighOnly0.16320.28770.02570.13800.77970.8796
Table 1: Ablation Study of DBSG-PINN on the 1D Multimodal Wave Equation (single seed)
ModelRelative L2 ↓\downarrowL∞L_{\infty} Error ↓\downarrowMean PDE Residual ↓\downarrowSpectral Error ↓\downarrowHF Recovery ↑\uparrowLF Recovery ↑\uparrow
DBSG-PINN (Full)0.10380.74160.01070.07000.79850.9459
NoGate0.58921.89050.01830.35710.58980.7446
LowOnly0.11080.75470.00840.07340.79330.9430
HighOnly0.67171.89970.05190.41160.54720.7174
Table 2: Ablation Study of DBSG-PINN on the Allen–Cahn Equation (single seed)
ModelRelative L2 ↓\downarrowL∞L_{\infty} Error ↓\downarrowMean PDE Residual ↓\downarrowSpectral Error ↓\downarrowHF Recovery ↑\uparrowLF Recovery ↑\uparrow
DBSG-PINN (Full)0.023980.167200.268100.014470.943870.99098
NoGate0.023830.159840.268360.014950.940790.99107
LowOnly0.028160.179970.277160.015930.935850.99089
HighOnly0.024550.163130.268660.016310.937670.99081
Table 3: Ablation Study of DBSG-PINN on the 1D Burgers Equation (single seed)
ModelRelative L2 Error ↓\downarrowL∞L_{\infty} Error ↓\downarrowMean PDE Residual ↓\downarrowSpectral Error ↓\downarrowHF Recovery ↑\uparrowLF Recovery ↑\uparrow
DBSG-PINN (Full)2.24×10−4\bm{2.24\times 10^{-4}}6.30×10−3\bm{6.30\times 10^{-3}}2.30×10−3\bm{2.30\times 10^{-3}}2.63×10−4\bm{2.63\times 10^{-4}}0.99560.9998
NoGate3.25×10−43.25\times 10^{-4}6.95×10−36.95\times 10^{-3}2.51×10−32.51\times 10^{-3}3.44×10−43.44\times 10^{-4}0.99800.9997
LowOnly4.77×10−44.77\times 10^{-4}6.30×10−36.30\times 10^{-3}2.53×10−32.53\times 10^{-3}5.37×10−45.37\times 10^{-4}0.99850.9995
HighOnly1.01×10−31.01\times 10^{-3}8.85×10−38.85\times 10^{-3}5.00×10−35.00\times 10^{-3}1.18×10−31.18\times 10^{-3}0.99370.9988
Table 4: Ablation Study of DBSG-PINN on the 1D Reaction–Diffusion Equation (single seed)
ModelRelative L2 ↓\downarrowL∞L_{\infty} Error ↓\downarrowMean PDE Residual ↓\downarrowSpectral Error ↓\downarrowHF Recovery ↑\uparrowLF Recovery ↑\uparrow
DBSG-PINN (Full)0.051340.083200.029310.026990.635390.97581
NoGate0.023970.051430.027110.008500.784240.99326
LowOnly0.491700.673220.022400.359380.299610.73716
HighOnly0.025140.048970.033400.015650.988910.98475
Table 5: Ablation Study of DBSG-PINN on the 1D Wave Equation (single seed)
Figure 2: Relative L2L_{2} error by ablation variant across the five benchmarks (single seed per bar). DBSG-PINN (Full) leads on Multimodal Wave, Allen–Cahn, and Reaction–Diffusion, while NoGate leads on Burgers and 1D Wave.
Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

HyperparameterBurgersReaction–Diff.Allen–CahnMultimodal Wave1D Wave
Low-frequency branch
Activationtanhtanhtanhtanhtanh
Hidden layers33433
Width2432643264
High-frequency branch
Activationsin⁡(ωx)\sin(\omega x)sin⁡(ωx)\sin(\omega x)sin⁡(ωx)\sin(\omega x)sin⁡(ωx)\sin(\omega x)sin⁡(ωx)\sin(\omega x)
Appendix
Table 6: Hyperparameter configuration of DBSG-PINN across the five benchmark PDEs. All entries use a single random seed per benchmark; see Section 4 for the resulting limitation on statistical confidence.
Figure 3: 1D Multimodal Wave, Full DBSG-PINN: exact solution, prediction, and absolute error.
Figure 4: 1D Multimodal Wave, HighOnly variant.
Figure 5: 1D Multimodal Wave, LowOnly variant.
Figure 6: 1D Multimodal Wave, NoGate variant.
Figure 7: 1D Multimodal Wave spectral recovery across ablation variants (FFT amplitude vs. frequency mode, single seed).
Figure 8: 1D Burgers Equation, Full DBSG-PINN: high-resolution reference solution, prediction, and absolute error.
Figure 9: 1D Burgers Equation, HighOnly variant.
Figure 10: 1D Burgers Equation, LowOnly variant.
Figure 11: 1D Burgers Equation, NoGate variant.
Figure 12: 1D Burgers Equation spectral recovery across ablation variants (FFT amplitude vs. frequency mode, single seed).
Figure 13: 1D Allen–Cahn Equation, Full DBSG-PINN: high-resolution reference solution, prediction, and absolute error.
Figure 14: 1D Allen–Cahn Equation, HighOnly variant.
Figure 15: 1D Allen–Cahn Equation, LowOnly variant.
Figure 16: 1D Allen–Cahn Equation, NoGate variant.
Figure 17: 1D Allen–Cahn Equation spectral recovery across ablation variants (FFT amplitude vs. frequency mode, single seed).
Figure 18: 1D Reaction–Diffusion Equation, Full DBSG-PINN: exact solution, prediction, and absolute error.
Figure 19: 1D Reaction–Diffusion Equation, HighOnly variant.
Figure 20: 1D Reaction–Diffusion Equation, LowOnly variant.
Figure 21: 1D Reaction–Diffusion Equation, NoGate variant.
Figure 22: 1D Reaction–Diffusion Equation spectral recovery across ablation variants (FFT amplitude vs. frequency mode, single seed).
Figure 23: 1D Wave Equation, Full DBSG-PINN: exact solution, prediction, and absolute error.
Figure 24: 1D Wave Equation, HighOnly variant.
Figure 25: 1D Wave Equation, LowOnly variant.
Figure 26: 1D Wave Equation, NoGate variant.
Figure 27: 1D Wave Equation spectral recovery across ablation variants (FFT amplitude vs. frequency mode, single seed).
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