Efficient Discrete Physics-informed Neural Networks for Addressing Evolutionary Partial Differential Equations

• URL: http://arxiv.org/abs/2312.14608v1
• Date: Fri, 22 Dec 2023 11:09:01 GMT
• Title: Efficient Discrete Physics-informed Neural Networks for Addressing Evolutionary Partial Differential Equations
• Authors: Siqi Chen, Bin Shan, Ye Li
• Abstract summary: Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. PINNs may violate the temporal causality property since all the temporal features in the PINNs loss are trained simultaneously. This paper proposes to use implicit time differencing schemes to enforce temporal causality, and use transfer learning to sequentially update the PINNs in space as surrogates for PDE solutions in different time frames.
• Score: 7.235476098729406
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. However, PINNs face training difficulties for evolutionary PDEs, particularly for dynamical systems whose solutions exhibit multi-scale or turbulent behavior over time. The reason is that PINNs may violate the temporal causality property since all the temporal features in the PINNs loss are trained simultaneously. This paper proposes to use implicit time differencing schemes to enforce temporal causality, and use transfer learning to sequentially update the PINNs in space as surrogates for PDE solutions in different time frames. The evolving PINNs are better able to capture the varying complexities of the evolutionary equations, while only requiring minor updates between adjacent time frames. Our method is theoretically proven to be convergent if the time step is small and each PINN in different time frames is well-trained. In addition, we provide state-of-the-art (SOTA) numerical results for a variety of benchmarks for which existing PINNs formulations may fail or be inefficient. We demonstrate that the proposed method improves the accuracy of PINNs approximation for evolutionary PDEs and improves efficiency by a factor of 4-40x.

Related papers

• A Sequential Meta-Transfer (SMT) Learning to Combat Complexities of Physics-Informed Neural Networks: Application to Composites Autoclave Processing [1.6317061277457001]
  PINNs have gained popularity in solving nonlinear partial differential equations. PINNs are designed to approximate a specific realization of a given PDE system. They lack the necessary generalizability to efficiently adapt to new system configurations.
  arXiv  Detail & Related papers  (2023-08-12T02:46:54Z)
• PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks [22.39904196850583]
  Physics-Informed Neural Networks (PINNs) have emerged as a promising deep learning framework for approximating numerical solutions to partial differential equations (PDEs) We introduce a novel Transformer-based framework, termed PINNsFormer, designed to address this limitation. PINNsFormer achieves superior generalization ability and accuracy across various scenarios, including PINNs failure modes and high-dimensional PDEs.
  arXiv  Detail & Related papers  (2023-07-21T18:06:27Z)
• iPINNs: Incremental learning for Physics-informed neural networks [66.4795381419701]
  Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs) We propose incremental PINNs that can learn multiple tasks sequentially without additional parameters for new tasks and improve performance for every equation in the sequence. Our approach learns multiple PDEs starting from the simplest one by creating its own subnetwork for each PDE and allowing each subnetwork to overlap with previously learnedworks.
  arXiv  Detail & Related papers  (2023-04-10T20:19:20Z)
• A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions [22.514769448363754]
  Training challenges in PINNs and XPINNs for time-dependent PDEs are discussed. We propose a new stacked-decomposition method that bridges the gap between PINNs and XPINNs. We also formulate a new time-sweeping collocation point algorithm inspired by the previous PINNs causality.
  arXiv  Detail & Related papers  (2023-02-28T01:19:21Z)
• PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers [4.1173475271436155]
  We propose a new kind of data-driven PDEs solver, physics-informed cell representations (PIXEL) PIXEL elegantly combines classical numerical methods and learning-based approaches. We show that PIXEL achieves fast convergence speed and high accuracy.
  arXiv  Detail & Related papers  (2022-07-26T10:46:56Z)
• Enforcing Continuous Physical Symmetries in Deep Learning Network for Solving Partial Differential Equations [3.6317085868198467]
  We introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries of PDEs are embedded into the loss function of PINN. We show that SPINN performs better than PINN with fewer training points and simpler architecture of neural network.
  arXiv  Detail & Related papers  (2022-06-19T00:44:22Z)
• Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture [77.59766598165551]
  Physics-informed neural networks (PINNs) are revolutionizing science and engineering practice by bringing together the power of deep learning to bear on scientific computation. Here, we propose Auto-PINN, which employs Neural Architecture Search (NAS) techniques to PINN design. A comprehensive set of pre-experiments using standard PDE benchmarks allows us to probe the structure-performance relationship in PINNs.
  arXiv  Detail & Related papers  (2022-05-27T03:24:31Z)
• Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method [70.19159220248805]
  Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs) We propose the generative adversarial neural network (GA-PINN), which integrates the generative adversarial (GA) mechanism with the structure of PINNs. Inspired from the weighting strategy of the Adaboost method, we then introduce a point-weighting (PW) method to improve the training efficiency of PINNs.
  arXiv  Detail & Related papers  (2022-05-18T06:50:44Z)
• Improved Training of Physics-Informed Neural Networks with Model Ensembles [81.38804205212425]
  We propose to expand the solution interval gradually to make the PINN converge to the correct solution. All ensemble members converge to the same solution in the vicinity of observed data. We show experimentally that the proposed method can improve the accuracy of the found solution.
  arXiv  Detail & Related papers  (2022-04-11T14:05:34Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Characterizing possible failure modes in physics-informed neural networks [55.83255669840384]
  Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs. We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
  arXiv  Detail & Related papers  (2021-09-02T16:06:45Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.