DeepONet在求解含参热传导方程中的应用

Application of DeepONet in Solving Heat Conduction Equation with Parameters

摘要: 含参热传导方程在多个科学与工程问题中广泛存在，由于所含参数具有随机性，热传导方程的解也随着不确定性的传播而具有随机性。为了刻画解的随机性，需要对该方程进行多次模拟仿真实验。另一方面，在实际应用中，我们需要在给定稀疏观测值的条件下，基于前向模型来识别方程中的未知参数。贝叶斯方法是识别未知参数的一个有效方法，借助观测数据，可以达到减小该方程中不确定性的目的。然而，参数和前向模型之间的非线性性导致后验分布没有显式表达式，我们拟利用构造蒙特卡罗马尔科夫链的方式，来实现后验分布的抽样。马尔科夫链的收敛要求数以百万次前向模型的仿真，构造前向模型的替代模型是提高抽样效率的有效方式之一。随着未知参数维数的增加，构建替代模型所需要的离线模拟次数也会呈指数级增加，这给参数识别问题带来了挑战。近年来，基于深度学习的算子学习方法，特别是DeepONet，为求解此类问题提供了新的思路。本文以DeepONet为核心工具，基于方程的部分解数据，构建和训练了一个随机热传导系数与观测数据之间的算子映射，并以此作为替代模型用于热传导方程的参数识别问题当中，提高了参数反演的效率。我们通过数值算例验证了DeepONet在求解含参偏微分方程时具有较高的精度，在高维未知参数的反演当中具有很好的应用效果。

Abstract: The parameterized heat conduction equation is widely present in many scientific and engineering problems. Since the parameters involved are random, the solution of the heat conduction equation is also random as the uncertainty propagates. In order to characterize the randomness of the solution, it is necessary to conduct multiple simulation experiments on the equation. On the other hand, in practical applications, we need to identify the unknown parameters in the equation based on the front-end model under the condition of given sparse observations. The Bayesian method is an effective method to identify unknown parameters. With the help of observed data, the uncertainty in the equation can be reduced. However, the nonlinearity between the parameters and the forward model leads to the lack of explicit expression of the posterior distribution. We intend to use the method of constructing Monte Carlo Markov chains to achieve sampling of the posterior distribution. The convergence of the Markov chain requires millions of simulations of the forward model. Constructing a surrogate model of the forward model is one of the effective ways to improve sampling efficiency. As the dimension of the unknown parameters increases, the number of offline simulations required to construct the surrogate model will also increase exponentially, which brings challenges to the parameter identification problem. In recent years, operator learning methods based on deep learning, especially DeepONet, have provided new ideas for solving such problems. This paper uses DeepONet as the core tool, constructs and trains an operator mapping between a random heat conduction coefficient and observation data based on partial solution data of the equation, and uses this as a substitute model for parameter identification of the heat conduction equation, improving the efficiency of parameter inversion. We verify through numerical examples that DeepONet has high accuracy in solving parameter-containing partial differential equations, and has a good application effect in the inversion of high-dimensional unknown parameters.