Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs). However, PINNs’ loss, the regularization terms, can only guarantee that the prediction results conform to the physical constraints in the average sense, which results in PINNs’ inability to strictly adhere to implied physical laws such as conservation laws and symmetries. This limits the optimization speed and accuracy of PINNs. Although some feature-enhanced PINNs attempt to address this issue by adding explicit constraints, their generality is limited due to specific question settings. To overcome this limitation, our study proposes the adaptive symmetry-recomposition PINN (AsPINN). By analyzing the parameter-sharing patterns of fully connected PINNs, specific network structures are developed to provide predictions with strict symmetry constraints. These structures are incorporated into diverse subnetworks to provide constrained intermediate outputs, then a specialized multi-head attention mechanism is attached to evaluate and composite them into final predictions adaptively. Thus, AsPINN maintains precise constraints while addressing the inability of individual structural subnetworks’ generality. This method is then applied to address several physically significant PDEs, including both forward and inverse problems. The numerical results demonstrates AsPINN’s mathematical consistency and generality, offering advantages in terms of optimization speed and accuracy with a reduced number of trainable parameters. The results also manifest that AsPINN mitigates the impact of ill-conditioned data.

中文翻译：

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AsPINN：自适应对称重组物理信息神经网络


物理信息神经网络 (PINN) 已显示出解决偏微分方程 (PDE) 的前景。然而，PINNs的损失，即正则化项，只能保证预测结果符合平均意义上的物理约束，这导致PINNs无法严格遵循守恒定律和对称性等隐含物理定律。这限制了 PINN 的优化速度和准确性。尽管一些功能增强的 PINN 试图通过添加显式约束来解决这个问题，但由于特定的问题设置，它们的通用性受到限制。为了克服这个限制，我们的研究提出了自适应对称重组 PINN (AsPINN)。通过分析全连接 PINN 的参数共享模式，开发了特定的网络结构来提供具有严格对称约束的预测。这些结构被纳入不同的子网络中，以提供受约束的中间输出，然后附加专门的多头注意机制来自适应地评估它们并将它们组合成最终预测。因此，AsPINN 保持了精确的约束，同时解决了单个结构子网络无法通用的问题。然后将该方法应用于解决几个物理上重要的偏微分方程，包括正向和逆向问题。数值结果证明了 AsPINN 的数学一致性和通用性，在优化速度和准确性方面具有优势，同时减少了可训练参数的数量。结果还表明 AsPINN 减轻了病态数据的影响。