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A Constrained Backpropagation Approach for the Adaptive Solution of Partial Differential Equations

机译:偏微分方程自适应解的约束反向传播方法

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This paper presents a constrained backpropagation (CPROP) methodology for solving nonlinear elliptic and parabolic partial differential equations (PDEs) adaptively, subject to changes in the PDE parameters or external forcing. Unlike existing methods based on penalty functions or Lagrange multipliers, CPROP solves the constrained optimization problem associated with training a neural network to approximate the PDE solution by means of direct elimination. As a result, CPROP reduces the dimensionality of the optimization problem, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic and parabolic PDEs with changing parameters and nonhomogeneous terms.
机译:本文提出了一种受约束的反向传播(CPROP)方法,该方法可以自适应地求解非线性椭圆和抛物线偏微分方程(PDE),这取决于PDE参数的变化或外部强迫。与现有的基于罚函数或拉格朗日乘子的方法不同,CPROP通过直接消除方法解决了与训练神经网络以逼近PDE解相关的约束优化问题。结果,在算法的每次迭代中,CPROP减少了优化问题的维数,同时精确地满足了与边界和初始条件相关的相等约束。通过几个示例证明了该方法的有效性,包括具有变化的参数和非齐次项的非线性椭圆和抛物线偏微分方程。

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