Recursive Regression with Neural Networks: Approximating the HJI PDE Solution
This work addresses a computational bottleneck in control theory for researchers dealing with high-dimensional systems, though it appears incremental as it builds on existing approximate dynamic programming with neural network integration.
The authors tackled the curse of dimensionality in approximating the Hamilton-Jacobi-Isaacs PDE by developing an approximate dynamic programming algorithm using neural networks, which reduced memory requirements compared to traditional gridding methods and was tested on systems up to six dimensions.
The majority of methods used to compute approximations to the Hamilton-Jacobi-Isaacs partial differential equation (HJI PDE) rely on the discretization of the state space to perform dynamic programming updates. This type of approach is known to suffer from the curse of dimensionality due to the exponential growth in grid points with the state dimension. In this work we present an approximate dynamic programming algorithm that computes an approximation of the solution of the HJI PDE by alternating between solving a regression problem and solving a minimax problem using a feedforward neural network as the function approximator. We find that this method requires less memory to run and to store the approximation than traditional gridding methods, and we test it on a few systems of two, three and six dimensions.