Fast and accurate predictions for complex physical dynamics are a significant challenge across various applications. Real-time prediction on resource-constrained hardware is even more crucial in real-world problems. The deep operator network (DeepONet) has recently been proposed as a framework for learning nonlinear mappings between function spaces. However, the DeepONet requires many parameters and has a high computational cost when learning operators, particularly those with complex (discontinuous or non-smooth) target functions. This study proposes HyperDeepONet, which uses the expressive power of the hypernetwork to enable the learning of a complex operator with a smaller set of parameters. The DeepONet and its variant models can be thought of as a method of injecting the input function information into the target function. From this perspective, these models can be viewed as a particular case of HyperDeepONet. We analyze the complexity of DeepONet and conclude that HyperDeepONet needs relatively lower complexity to obtain the desired accuracy for operator learning. HyperDeepONet successfully learned various operators with fewer computational resources compared to other benchmarks.

翻译：快速且准确预测复杂物理动力学是各类应用中的重大挑战。在资源受限硬件上实现实时预测对于现实问题更为关键。深度算子网络（DeepONet）近期被提出作为学习函数空间之间非线性映射的框架。然而，DeepONet在学习算子时（尤其是具有复杂（不连续或非光滑）目标函数的算子）需要大量参数且计算成本高昂。本研究提出HyperDeepONet，利用超网络的表达能力以更少的参数集实现复杂算子的学习。DeepONet及其变体模型可视为将输入函数信息注入目标函数的方法，从这一角度看，这些模型可被视作HyperDeepONet的特例。我们分析了DeepONet的复杂度，并得出结论：HyperDeepONet需要相对较低的复杂度即可获得期望的算子学习精度。相较于其他基准模型，HyperDeepONet以更少的计算资源成功学习了多种算子。