[论文解读] Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions

Bin Lin, Zhiping Mao+2 more|arXiv (Cornell University)|Oct 30, 2023

Model Reduction and Neural Networks被引用 6

一句话总结

该论文提出了一种算子学习增强的物理信息神经网络（OL-PINN），通过结合预训练的DeepONet以处理平滑解，同时利用PINN求解具有陡峭解的PDE。通过利用算子学习来引导PINN，OL-PINN在极少残差点的情况下实现了高精度与高稳定性，显著优于基线PINN，在高雷诺数Navier-Stokes方程和Burgers方程等复杂问题上表现更优。

ABSTRACT

Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile, the neural operator approach, including methods such as Deep Operator Network (DeepONet) and Fourier neural operator (FNO), has been introduced and extensively employed in approximating solution of PDEs. Nevertheless, to solve problems consisting of sharp solutions poses a significant challenge when employing these two approaches. To address this issue, we propose in this work a novel framework termed Operator Learning Enhanced Physics-informed Neural Networks (OL-PINN). Initially, we utilize DeepONet to learn the solution operator for a set of smooth problems relevant to the PDEs characterized by sharp solutions. Subsequently, we integrate the pre-trained DeepONet with PINN to resolve the target sharp solution problem. We showcase the efficacy of OL-PINN by successfully addressing various problems, such as the nonlinear diffusion-reaction equation, the Burgers equation and the incompressible Navier-Stokes equation at high Reynolds number. Compared with the vanilla PINN, the proposed method requires only a small number of residual points to achieve a strong generalization capability. Moreover, it substantially enhances accuracy, while also ensuring a robust training process. Furthermore, OL-PINN inherits the advantage of PINN for solving inverse problems. To this end, we apply the OL-PINN approach for solving problems with only partial boundary conditions, which usually cannot be solved by the classical numerical methods, showing its capacity in solving ill-posed problems and consequently more complex inverse problems.

研究动机与目标

• 解决基线PINN在求解具有陡峭解或不连续性的PDE时泛化能力差与不稳定的缺陷。
• 克服PINN中自适应采样与激活函数带来的计算成本高与不稳定的局限性。
• 实现对边界条件不完整等不适定逆问题的鲁棒求解，此类问题对经典数值方法构成挑战。
• 利用平滑粘性解与陡峭无粘性解之间的渐近关系，提升对奇异特征的学习能力。
• 构建一种混合框架，结合DeepONet（算子学习）与PINN（物理信息训练）的优势，实现高精度的PDE求解。

提出的方法

• 预训练一个DeepONet，以学习将初始条件映射到与目标陡峭解问题相关的平滑粘性PDE族解的解算子。
• 将预训练的DeepONet作为归纳偏置，嵌入PINN损失函数中，增强其捕捉陡峭跃迁的能力。
• 将DeepONet的输出作为先验或辅助头集成到PINN架构中，提升泛化能力，且无需重新训练整个网络。
• 将混合OL-PINN框架应用于正问题与逆问题，包括边界条件部分或不足的情况。
• 通过无散参数化方法，直接在神经网络架构中施加物理约束（如不可压缩性，$\nabla \cdot \mathbf{u} = 0$）。
• 采用自适应时空残差点采样（如$22 \times 22 \times 20$）与Adam优化器，训练50,000个周期，以高效训练模型。

实验结果

研究问题

• RQ1 通过DeepONet进行算子学习，能否提升PINN在求解具有陡峭解的PDE时的泛化能力与稳定性？
• RQ2 在高雷诺数Navier-Stokes方程与Burgers方程中，OL-PINN相较于基线PINN在精度与训练稳定性方面表现如何？
• RQ3 OL-PINN能否有效求解边界条件不完整等不适定逆问题，而传统方法在此类问题上失效？
• RQ4 在双曲型PDE的平滑粘性类比问题上预训练DeepONet，是否能提升对无粘性、激波捕捉解的学习能力？
• RQ5 OL-PINN在具有陡峭梯度的问题上，达到高精度所需的最少残差点数量是多少？

主要发现

• 在Case I中，OL-PINN将相对$L^2$测试误差降低至$2.21 \pm 0.73\%$，Case II中为$2.93 \pm 1.29\%$，而基线PINN的误差为$30.16 \pm 71.60\%$。
• 该方法仅使用$22 \times 22 \times 20$个残差点即实现高精度，表明其在极少监督下仍具备强大泛化能力。
• OL-PINN保持了稳定的训练过程，避免了自适应PINN变体或依赖高密度残差点方法中常见的数值不稳定性。
• 对于仅在涡度上具有周期性边界条件的不适定问题，OL-PINN实现$3.36 \pm 1.05\%$的误差，显著优于基线PINN的$43.28 \pm 16.52\%$。
• 该框架成功求解了高雷诺数下的不可压缩Navier-Stokes方程与顶盖驱动流问题，准确捕捉了复杂的涡旋结构。
• 预训练的DeepONet提供了强有力的归纳偏置，使PINN即使在未显式采样陡峭特征的情况下也能实现有效泛化。