Multi-task learning (MTL) is an inductive transfer mechanism designed to leverage useful information from multiple tasks to improve generalization performance compared to single-task learning. It has been extensively explored in traditional machine learning to address issues such as data sparsity and overfitting in neural networks. In this work, we apply MTL to problems in science and engineering governed by partial differential equations (PDEs). However, implementing MTL in this context is complex, as it requires task-specific modifications to accommodate various scenarios representing different physical processes. To this end, we present a multi-task deep operator network (MT-DeepONet) to learn solutions across various functional forms of source terms in a PDE and multiple geometries in a single concurrent training session. We introduce modifications in the branch network of the vanilla DeepONet to account for various functional forms of a parameterized coefficient in a PDE. Additionally, we handle parameterized geometries by introducing a binary mask in the branch network and incorporating it into the loss term to improve convergence and generalization to new geometry tasks. Our approach is demonstrated on three benchmark problems: (1) learning different functional forms of the source term in the Fisher equation; (2) learning multiple geometries in a 2D Darcy Flow problem and showcasing better transfer learning capabilities to new geometries; and (3) learning 3D parameterized geometries for a heat transfer problem and demonstrate the ability to predict on new but similar geometries. Our MT-DeepONet framework offers a novel approach to solving PDE problems in engineering and science under a unified umbrella based on synergistic learning that reduces the overall training cost for neural operators.

翻译：多任务学习（MTL）是一种归纳迁移机制，旨在通过利用多个任务中的有效信息来提升泛化性能，相较于单任务学习具有显著优势。在传统机器学习领域，MTL已被广泛探索，用于解决神经网络中的数据稀疏性和过拟合等问题。本研究将MTL应用于由偏微分方程（PDEs）描述的科学与工程问题。然而，在此背景下实施MTL较为复杂，因为它需要针对特定任务进行调整，以适应代表不同物理过程的多种场景。为此，我们提出了一种多任务深度算子网络（MT-DeepONet），能够在一次并行训练中学习PDE中不同源项函数形式以及多种几何构型的解。我们对基础DeepONet的分支网络进行了改进，以处理PDE中参数化系数的多种函数形式。此外，我们通过引入分支网络中的二元掩码并将其纳入损失项，来处理参数化几何问题，从而提升收敛性以及对新几何任务的泛化能力。我们在三个基准问题上验证了所提方法：（1）学习Fisher方程中源项的不同函数形式；（2）在二维达西流问题中学习多种几何构型，并展示其对新几何构型更优的迁移学习能力；（3）为传热问题学习三维参数化几何构型，并证明其能够对相似的新几何构型进行预测。我们的MT-DeepONet框架基于协同学习，为在统一框架下解决工程与科学中的PDE问题提供了一种新方法，显著降低了神经算子网络的总体训练成本。