Research map / Probabilistic and scientific modeling / Differential equations and PDEs
CoPINN: Cognitive Physics-Informed Neural Networks
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Proposes CoPINN, a separable-subnetwork PINN that weights collocation points by PDE-residual-gradient difficulty and schedules training from easy to hard via a cognitive (self-paced) scheduler, outperforming seven PINN baselines on PDE benchmarks.
Problem
Standard PINNs treat easy and hard collocation points equally, so errors concentrate near physical boundaries, causing unstable learning, poor local minima and an "Unbalanced Prediction Problem" (UPP).
Approach
Combines SPINN-style separable per-axis subnetworks with a self-paced learning scheme that measures each collocation point's difficulty by the gradient of the PDE residual and linearly shifts sample weights from easy to hard over training epochs.
Key results
- On Helmholtz with Nc=256^3, CoPINN reaches RL2 0.0006, a 98% improvement over the second-best method SPINN(m).
- CoPINN achieves best RL2 and RMSE across all Helmholtz, Diffusion and (2+1)-d Klein-Gordon settings, while several baselines run out of memory at 128^3 and 256^3 collocation points.
- On the 3+1-d Klein-Gordon equation CoPINN improves RL2 by 47% to 78% over the second-best method at smaller collocation counts.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CoPINN instead of SPINN (m) | Helmholtz · Helmholtz, Nc=323 | RL2: 0.0595 -> 0.0050 | Helped |
| Use CoPINN instead of SPINN (m) | (2+1)-d Klein-Gordon · (2+1)-d Klein-Gordon, Nc=323 | RL2: 0.0020 -> 0.0006 | Helped |
| Use CoPINN instead of SPINN (m) | (3+1)-d Klein-Gordon · (3+1)-d Klein-Gordon, Nc=323 | RL2: 0.0065 -> 0.0027 | Helped |
| Use CoPINN instead of FPINN | Diffusion · Diffusion, Nc=323 | RL2: 0.0048 -> 0.0038 | Helped |
| Use CoPINN instead of SPINN (m) | Flow-mixing 3D · Flow-mixing 3D, Nc=323 | RL2: 0.0263 -> 0.0230 | Helped |
| Apply CoPINN difficulty evaluation and cognitive training scheduler to PINN | PINN · Convection, ξ=30 and ξ=50 | RL2: ξ=30: 0.0160 -> 0.0055; ξ=50: 0.5471 -> 0.5811 | Mixed |
| Add cognitive training scheduler instead of using the original loss function (ablation) | CoPINN · (2+1)-d Klein-Gordon, Nc=323 | RL2: 0.0020 -> 0.0006 | Helped |
| Add the epoch change component to the scheduler (ablation) | CoPINN without epoch change component · (2+1)-d Klein-Gordon, Nc=323 | RL2: 0.0027 -> 0.0006 | Helped |
| Add the sample change component to the scheduler (ablation) | CoPINN without sample change component · (2+1)-d Klein-Gordon, Nc=323 | RL2: 0.0010 -> 0.0006 | Helped |
Surprising findings
- The paper reports that the epoch-wise weight change component matters more than the per-sample difficulty component in the ablation.
- On the 1D Convection equation with xi=50, CoPINN (0.5811) is not best, trailing Causal R3 (0.5471).
Limitations
- Up-weighting hard samples may cause overfitting to difficult points and neglect others.
- Dynamic sample weighting can introduce training instability when hard-sample weights grow large.
- Results are on manufactured or simulated PDE benchmarks with limited hyperparameter sensitivity (beta) requiring tuning.
Experimental setup
Datasets: Helmholtz equation (manufactured solution) Diffusion equation (FEniCS reference, 101^3) (2+1)-d Klein-Gordon (3+1)-d Klein-Gordon (2+1)-d Flow Mixing 1D Convection equation
Benchmarks: Helmholtz Diffusion (2+1)-d Klein-Gordon (3+1)-d Klein-Gordon Flow-mixing 3D 1D Convection
Metrics: Relative L2 error RMSE improvement percentage
Baselines: PINN gPINN SPINN SPINN(m) AHD-PINN RoPINN FPINN
Builds on: PINN SPINN self-paced learning gPINN RoPINN
Closest papers by meaning
- ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks (ICLR 2025 · similarity 0.88)
- Parameterized Physics-informed Neural Networks for Parameterized PDEs (ICML 2024 · similarity 0.88)
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.88)
- RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.88)
- Neuro-Spectral Architectures for Causal Physics-Informed Networks (NeurIPS 2025 · similarity 0.87)
- Dual Cone Gradient Descent for Training Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.87)
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