标题： 随机算子网络：一种基于随机最大值原理的算子学习方法 显示英文标题

标题： Stochastic Operator Network: A Stochastic Maximum Principle Based Approach to Operator Learning

摘要： 我们开发了一个用于算子学习中不确定性量化的新框架，即随机算子网络（SON）。 SON 将随机神经网络（SNN）的随机最优控制概念与 DeepONet 结合起来。 通过将分支网络表述为随机微分方程并通过对偶倒向随机微分方程进行反向传播，我们在 SGD 更新中用随机最大原理中的哈密顿量梯度替换了损失函数的梯度。 这使得 SON 能够通过其扩散参数学习算子中存在的不确定性。 然后，我们在 2D 和 3D 中演示了 SON 在复制多个噪声算子时的有效性。 显示英文摘要

摘要： We develop a novel framework for uncertainty quantification in operator learning, the Stochastic Operator Network (SON). SON combines the stochastic optimal control concepts of the Stochastic Neural Network (SNN) with the DeepONet. By formulating the branch net as an SDE and backpropagating through the adjoint BSDE, we replace the gradient of the loss function with the gradient of the Hamiltonian from Stohastic Maximum Principle in the SGD update. This allows SON to learn the uncertainty present in operators through its diffusion parameters. We then demonstrate the effectiveness of SON when replicating several noisy operators in 2D and 3D.