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Neuro-Spectral Architectures for Causal Physics-Informed Networks
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Proposes NeuSA, a PINN architecture that projects a PDE onto a Fourier/sine-cosine spectral basis and integrates the coefficients with a Neural ODE, initialized from a linear spectral multiplier, to solve wave, sine-Gordon and Burgers' equations with lower error and faster training than MLP-based PINNs.
Problem
Standard MLP-based PINNs struggle to converge on time-dependent initial value problems (wave, nonlinear wave, Burgers' equations), violating causality and suffering from spectral bias toward low frequencies.
Approach
Combine a spectral (Fourier/sine/cosine) basis expansion of the solution with a Neural ODE that evolves the spectral coefficients in time, initialized as the analytical Fourier multiplier of the linearized PDE plus a small learned correction; this gives exact initial conditions, flow-based causality, and spectral high-frequency representation.
Key results
- On the 2D Layers wave equation NeuSA reaches rMSE 0.075 vs 0.115 for QRes (best baseline) with 530 s training time.
- On sine-Gordon NeuSA reaches rMSE 0.001 vs 0.020 for QRes, trained in 215 s vs 1,315 s.
- On 2D Burgers' NeuSA reaches rMSE 0.051 in 112 s vs 0.073 for QRes at 1,135 s.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace QRes with NeuSA | PINN architecture · 2D Layers | rMSE: 0.115 -> 0.075 | Helped |
| replace QRes with NeuSA | PINN architecture · 2D Layers | training time (s): 750 -> 530 | Helped |
| replace QRes with NeuSA | PINN architecture · Marmousi | rMSE: 0.412 -> 0.171 | Helped |
| replace QRes with NeuSA | PINN architecture · Marmousi | training time (s): 718 -> 573 | Helped |
| replace QRes with NeuSA | PINN architecture · 3D Layers | rMSE: 0.021 -> 0.008 | Helped |
| replace QRes with NeuSA | PINN architecture · 3D Layers | training time (s): 8775 -> 702 | Helped |
| replace QRes with NeuSA | PINN architecture · 2D Burgers | rMSE: 0.073 -> 0.051 | Helped |
| replace QRes with NeuSA | PINN architecture · 2D Burgers | training time (s): 1,135 -> 112 | Helped |
| replace QRes with NeuSA | PINN architecture · Sine Gordon | rMSE: 0.020 -> 0.001 | Helped |
| replace QRes with NeuSA | PINN architecture · Sine Gordon | training time (s): 1,315 -> 215 | Helped |
Surprising findings
- Baselines degrade when trained with learning rate 1e-2 due to instability, whereas NeuSA benefits from larger learning rates.
- NeuSA's performance is strongly influenced by analytical initialization and diminishes when no prior dynamical information is available.
Limitations
- Restricted to simple rectangular domains where Fourier/sine/cosine bases apply; complex geometries need other bases.
- Fourth-order Runge-Kutta integration may be unstable on stiff problems.
- Performance depends strongly on the analytical linear initialization and degrades without prior dynamical information.
Experimental setup
Datasets: Marmousi model (2D heterogeneous medium) synthetic wave/Burgers/sine-Gordon problems with numerical ground truth
Benchmarks: 2D Layers wave equation Marmousi wave equation 3D Layers wave equation 2D Burgers' equation 1D sine-Gordon equation
Metrics: rMSE rMAE training time (s)
Baselines: PINN QRes FLS PINNsFormer
Builds on: PINN Neural ODE Fourier feature networks PINNsFormer QRes
Closest papers by meaning
- Neural Spectral Methods: Self-supervised learning in the spectral domain (ICLR 2024 · similarity 0.89)
- HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions (NeurIPS 2025 · similarity 0.88)
- Efficient Error Certification for Physics-Informed Neural Networks (ICML 2024 · similarity 0.88)
- CoPINN: Cognitive Physics-Informed Neural Networks (ICML 2025 · similarity 0.87)
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.87)
- Sub-Sequential Physics-Informed Learning with State Space Model (ICML 2025 · similarity 0.87)
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