Neural Operator Networks (ONets) represent a novel advancement in machine learning algorithms, offering a robust and generalizable alternative for approximating partial differential equations (PDEs) solutions. Unlike traditional Neural Networks (NN), which directly approximate functions, ONets specialize in approximating mathematical operators, enhancing their efficacy in addressing complex PDEs. In this work, we evaluate the capabilities of Deep Operator Networks (DeepONets), an ONets implementation using a branch/trunk architecture. Three test cases are studied: a system of ODEs, a general diffusion system, and the convection/diffusion Burgers equation. It is demonstrated that DeepONets can accurately learn the solution operators, achieving prediction accuracy scores above 0.96 for the ODE and diffusion problems over the observed domain while achieving zero shot (without retraining) capability. More importantly, when evaluated on unseen scenarios (zero shot feature), the trained models exhibit excellent generalization ability. This underscores ONets vital niche for surrogate modeling and digital twin development across physical systems. While convection-diffusion poses a greater challenge, the results confirm the promise of ONets and motivate further enhancements to the DeepONet algorithm. This work represents an important step towards unlocking the potential of digital twins through robust and generalizable surrogates.

翻译：神经算子网络（ONets）代表了机器学习算法的一项前沿进展，为逼近偏微分方程（PDE）解提供了稳健且具备泛化能力的替代方案。传统神经网络（NN）直接逼近函数，而ONets专注于逼近数学算子，从而提升其处理复杂PDE的有效性。本研究评估了采用分支/主干架构的ONets实现——深度算子网络（DeepONets）的能力。我们研究了三个测试案例：一个常微分方程组、一个通用扩散系统以及对流/扩散Burgers方程。结果表明，DeepONets能够准确学习解算子，在观测域内对ODE和扩散问题的预测准确率得分超过0.96，并具备零样本（无需重新训练）能力。更重要的是，在未见场景（零样本特性）下评估时，训练后的模型展现出优异的泛化能力。这凸显了ONets在跨物理系统的代理建模和数字孪生开发中的关键作用。尽管对流/扩散问题构成更大挑战，但结果证实了ONets的潜力，并推动对DeepONet算法的进一步改进。本研究是迈向通过稳健且具泛化能力的代理模型释放数字孪生潜力的重要一步。