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Solving Coupled Differential Equation Groups Using PINO-CDE

arXiv 2210.00222 ↗First posted 1 Oct 2022cs.LGcited by 0

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Abstract (arXiv)

As a fundamental mathmatical tool in many engineering disciplines, coupled differential equation groups are being widely used to model complex structures containing multiple physical quantities. Engineers constantly adjust structural parameters at the design stage, which requires a highly efficient solver. The rise of deep learning technologies has offered new perspectives on this task. Unfortunately, existing black-box models suffer from poor accuracy and robustness, while the advanced methodologies of single-output operator regression cannot deal with multiple quantities simultaneously. To address these challenges, we propose PINO-CDE, a deep learning framework for solving coupled differential equation groups (CDEs) along with an equation normalization algorithm for performance enhancing. Based on the theory of physics-informed neural operator (PINO), PINO-CDE uses a single network for all quantities in a CDEs, instead of training dozens, or even hundreds of networks as in the existing literature. We demonstrate the flexibility and feasibility of PINO-CDE for one toy example and two engineering applications: vehicle-track coupled dynamics (VTCD) and reliability assessment for a four-storey building (uncertainty propagation). The performance of VTCD indicates that PINO-CDE outperforms existing software and deep learning-based methods in terms of efficiency and precision, respectively. For the uncertainty propagation task, PINO-CDE provides higher-resolution results in less than a quarter of the time incurred when using the probability density evolution method (PDEM). This framework integrates engineering dynamics and deep learning technologies and may reveal a new concept for CDEs solving and uncertainty propagation.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 19 of its 63 references that are in the map. First posted on arXiv 1 Oct 2022.

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References · 19

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2017GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networkscited by 816
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  5. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  6. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  7. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  8. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  9. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  10. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  11. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  12. 2017Solving ill-posed inverse problems using iterative deep neural networkscited by 145
7 more
  1. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  2. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  3. 2022A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problemscited by 20
  4. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  5. 2022Assessment of DeepONet for reliability analysis of stochastic nonlinear dynamical systemscited by 11
  6. 2019A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regimecited by 7
  7. 2019Data-driven simulation for general purpose multibody dynamics using deep neural networkscited by 6

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