基于多指标贝尔多项式计算物理信息神经网络中的高阶混合导数
Computing high-order mixed derivatives in physics-informed neural networks using multi-index Bell polynomials
AI总结：
该研究提出基于多指标贝尔多项式的方法，计算物理信息神经网络的高阶混合导数，避免内存故障，经7阶验证，可用于多种偏微分方程测试，7阶Zakharov-Kuznetsov测试相对误差达6×10⁻⁴。
AI中文摘要：
用于求解高阶偏微分方程的物理信息神经网络（Physics-informed neural networks, PINNs）需要混合输入导数及其相对于网络参数的梯度。标准实现通过重复自动微分获取K阶导数，而我们则对多元Faà di Bruno公式的前向递归及其在规定的下闭多指标集上的显式反向传播进行组织，其中贝尔多项式卷积仅制表一次。前向传播仅携带微分算子所需的导数；反向传播则从由这些导数的任意子集（包括非线性乘积和耦合场）构成的损失中传播梯度。两种递归均精确到舍入误差，且避免了嵌套计算图。通过独立的泰勒喷流（Taylor-jet）实现、测试问题的符号检查及有限差分法，验证了直至7阶的导数和损失梯度。在单个CPU核心上，该方法可计算关于4个输入的直至7阶的330个混合导数及相应的损失梯度，且不会出现若干嵌套实现所观察到的内存故障。数值测试包括3阶、5阶和7阶色散方程、不可压缩流动及人工构造的五场电流体动力学系统，其中3+1维的7阶Zakharov-Kuznetsov测试的相对解误差为6×10⁻⁴。
英文摘要：
Physics-informed neural networks for high-order partial differential equations require mixed input derivatives and their gradients with respect to the network parameters. Standard implementations obtain an order-$K$ derivative by repeated automatic differentiation. We instead organize the forward recursion of the multivariate Faà di Bruno formula and its explicit backpropagation over a prescribed downward-closed set of multi-indices, with the Bell-polynomial convolutions tabulated once. The forward pass carries only the derivatives the differential operator requires. The backward pass propagates gradients from losses formed from any subset of them, including nonlinear products and coupled fields. Both recursions are exact up to roundoff and avoid nested computational graphs. An independent Taylor-jet implementation, symbolic checks of the test problems, and finite differences verify derivatives and loss gradients through order seven. On one CPU core, the method evaluates 330 mixed derivatives with respect to four inputs through order seven and the corresponding loss gradient without the memory failures observed for several nested implementations. Numerical tests include third-, fifth-, and seventh-order dispersive equations, incompressible flow, and a manufactured five-field electrohydrodynamic system. The seventh-order Zakharov-Kuznetsov test in $3+1$ dimensions has a relative solution error of $6\times10^{-4}$.
