Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs). However, finding a set of neural network parameters that lead to fulfilling a PDE can be challenging and non-unique due to the complexity of the loss landscape that needs to be traversed. Although a variety of multi-task learning and transfer learning approaches have been proposed to overcome these issues, there is no incremental training procedure for PINNs that can effectively mitigate such training challenges. We propose incremental PINNs (iPINNs) that can learn multiple tasks (equations) 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 learned subnetworks. We demonstrate that previous subnetworks are a good initialization for a new equation if PDEs share similarities. We also show that iPINNs achieve lower prediction error than regular PINNs for two different scenarios: (1) learning a family of equations (e.g., 1-D convection PDE); and (2) learning PDEs resulting from a combination of processes (e.g., 1-D reaction-diffusion PDE). The ability to learn all problems with a single network together with learning more complex PDEs with better generalization than regular PINNs will open new avenues in this field.

翻译：物理信息神经网络（PINNs）近期已成为求解偏微分方程（PDEs）的有力工具。然而，由于需要穿越的损失景观的复杂性，找到一组能够满足PDE的神经网络参数往往具有挑战性且结果不唯一。尽管已提出多种多任务学习和迁移学习方法来解决这些问题，但目前尚无针对PINNs的增量训练流程来有效缓解此类训练挑战。我们提出增量PINNs（iPINNs），该方法无需为新任务增加额外参数即可顺序学习多个任务（方程），并提升序列中每个方程的性能。我们的方法从最简单的PDE开始学习，通过为每个PDE创建其专属子网络，并允许每个子网络与先前学习的子网络重叠。我们证明，若PDE具有相似性，先前的子网络可为新方程提供良好的初始化。我们还展示iPINNs在两种不同场景下比常规PINNs实现更低的预测误差：（1）学习一系列方程（如一维对流PDE）；（2）学习由多个过程组合形成的PDE（如一维反应-扩散PDE）。相较于常规PINNs，iPINNs能以单一网络学习所有问题，并对更复杂PDE实现更优泛化能力，这将为该领域开辟新途径。