CoPINN: Cognitive Physics-Informed Neural Networks

Siyuan Duan, Wenyuan Wu, Peng Hu, Zhenwen Ren, Dezhong Peng, Yuan Sun
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:14759-14777, 2025.

Abstract

Physics-informed neural networks (PINNs) aim to constrain the outputs and gradients of deep learning models to satisfy specified governing physics equations, which have demonstrated significant potential for solving partial differential equations (PDEs). Although existing PINN methods have achieved pleasing performance, they always treat both easy and hard sample points indiscriminately, especially ones in the physical boundaries. This easily causes the PINN model to fall into undesirable local minima and unstable learning, thereby resulting in an Unbalanced Prediction Problem (UPP). To deal with this daunting problem, we propose a novel framework named Cognitive Physical Informed Neural Network (CoPINN) that imitates the human cognitive learning manner from easy to hard. Specifically, we first employ separable subnetworks to encode independent one-dimensional coordinates and apply an aggregation scheme to generate multi-dimensional predicted physical variables. Then, during the training phase, we dynamically evaluate the difficulty of each sample according to the gradient of the PDE residuals. Finally, we propose a cognitive training scheduler to progressively optimize the entire sampling regions from easy to hard, thereby embracing robustness and generalization against predicting physical boundary regions. Extensive experiments demonstrate that our CoPINN achieves state-of-the-art performance, particularly significantly reducing prediction errors in stubborn regions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-duan25b, title = {{C}o{PINN}: Cognitive Physics-Informed Neural Networks}, author = {Duan, Siyuan and Wu, Wenyuan and Hu, Peng and Ren, Zhenwen and Peng, Dezhong and Sun, Yuan}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {14759--14777}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/duan25b/duan25b.pdf}, url = {https://proceedings.mlr.press/v267/duan25b.html}, abstract = {Physics-informed neural networks (PINNs) aim to constrain the outputs and gradients of deep learning models to satisfy specified governing physics equations, which have demonstrated significant potential for solving partial differential equations (PDEs). Although existing PINN methods have achieved pleasing performance, they always treat both easy and hard sample points indiscriminately, especially ones in the physical boundaries. This easily causes the PINN model to fall into undesirable local minima and unstable learning, thereby resulting in an Unbalanced Prediction Problem (UPP). To deal with this daunting problem, we propose a novel framework named Cognitive Physical Informed Neural Network (CoPINN) that imitates the human cognitive learning manner from easy to hard. Specifically, we first employ separable subnetworks to encode independent one-dimensional coordinates and apply an aggregation scheme to generate multi-dimensional predicted physical variables. Then, during the training phase, we dynamically evaluate the difficulty of each sample according to the gradient of the PDE residuals. Finally, we propose a cognitive training scheduler to progressively optimize the entire sampling regions from easy to hard, thereby embracing robustness and generalization against predicting physical boundary regions. Extensive experiments demonstrate that our CoPINN achieves state-of-the-art performance, particularly significantly reducing prediction errors in stubborn regions.} }
Endnote
%0 Conference Paper %T CoPINN: Cognitive Physics-Informed Neural Networks %A Siyuan Duan %A Wenyuan Wu %A Peng Hu %A Zhenwen Ren %A Dezhong Peng %A Yuan Sun %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-duan25b %I PMLR %P 14759--14777 %U https://proceedings.mlr.press/v267/duan25b.html %V 267 %X Physics-informed neural networks (PINNs) aim to constrain the outputs and gradients of deep learning models to satisfy specified governing physics equations, which have demonstrated significant potential for solving partial differential equations (PDEs). Although existing PINN methods have achieved pleasing performance, they always treat both easy and hard sample points indiscriminately, especially ones in the physical boundaries. This easily causes the PINN model to fall into undesirable local minima and unstable learning, thereby resulting in an Unbalanced Prediction Problem (UPP). To deal with this daunting problem, we propose a novel framework named Cognitive Physical Informed Neural Network (CoPINN) that imitates the human cognitive learning manner from easy to hard. Specifically, we first employ separable subnetworks to encode independent one-dimensional coordinates and apply an aggregation scheme to generate multi-dimensional predicted physical variables. Then, during the training phase, we dynamically evaluate the difficulty of each sample according to the gradient of the PDE residuals. Finally, we propose a cognitive training scheduler to progressively optimize the entire sampling regions from easy to hard, thereby embracing robustness and generalization against predicting physical boundary regions. Extensive experiments demonstrate that our CoPINN achieves state-of-the-art performance, particularly significantly reducing prediction errors in stubborn regions.
APA
Duan, S., Wu, W., Hu, P., Ren, Z., Peng, D. & Sun, Y.. (2025). CoPINN: Cognitive Physics-Informed Neural Networks. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:14759-14777 Available from https://proceedings.mlr.press/v267/duan25b.html.

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