非凸区域微分方程的神经网络解法--求解狭缝阱微流控器件中的电场
Neural Network Solutions to Differential Equations in Non-Convex Domains: Solving the Electric Field in the Slit-Well Microfluidic Device
摘要：
采用求解微分方程的神经网络方法对狭缝阱微流控装置中的电势和相应的电场进行了近似计算。该设备的几何形状是非凸的，这使得使用神经网络方法解决这一问题具有挑战性。为了验证该方法，神经网络的解决方案相比，使用有限元方法获得的参考解决方案。提出了其他指标来衡量神经网络恢复训练期间未明确实施的重要物理不变量的能力：空间对称性和电通量守恒。最后，作为一个特定应用程序的有效性测试，神经网络电场被纳入粒子模拟。方便的是，用于训练神经网络的相同损失泛函似乎也提供了网络真实误差的可靠估计，如本文所考虑的任何度量所测量的。在所有指标中，深度神经网络的表现都明显优于浅层神经网络，即使在按计算成本进行归一化时也是如此。总之，结果表明，神经网络方法可以可靠地产生可接受的精度的解决方案，用于后续的物理计算，如粒子模拟。
英文摘要：
The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging problem to solve using the neural network method. To validate the method, the neural network solutions are compared to a reference solution obtained using the finite element method. Additional metrics are presented that measure how well the neural networks recover important physical invariants that are not explicitly enforced during training: spatial symmetries and conservation of electric flux. Finally, as an application-specific test of validity, neural network electric fields are incorporated into particle simulations. Conveniently, the same loss functional used to train the neural networks also seems to provide a reliable estimator of the networks' true errors, as measured by any of the metrics considered here. In all metrics, deep neural networks significantly outperform shallow neural networks, even when normalized by computational cost. Altogether, the results suggest that the neural network method can reliably produce solutions of acceptable accuracy for use in subsequent physical computations, such as particle simulations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
