论文标题
深度神经网络克服了半连接部分微分方程的数值近似值的维度诅咒
Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations
论文作者
论文摘要
我们证明，在独立于渐变的，lipschitz的连续性非线性的情况下，深层神经网络能够近似半线性的kolmogorov pde解决方案，而网络中所需的参数数量在\ mathbb in \ mathbb {n} $ n} $ n and var n} $ n v pccice $ n v pccice $ n v pccice $ n and n dime and dimensial上生长。以前，仅在半线性热方程式的情况下才证明这一点。
We prove that deep neural networks are capable of approximating solutions of semilinear Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the required number of parameters in the networks grow at most polynomially in both dimension $d \in \mathbb{N}$ and prescribed reciprocal accuracy $\varepsilon$. Previously, this has only been proven in the case of semilinear heat equations.
