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This study introduces a numerical algorithm for approximating optimal controls in stochastic reaction-diffusion equations. The method reduces computational complexity, offering asymptotically optimal cost approximations using artificial neural networks and radial basis functions.

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Area of Science:

  • Numerical analysis
  • Stochastic control theory
  • Partial differential equations

Background:

  • Stochastic reaction-diffusion equations (SRDEs) are crucial in modeling complex systems with inherent randomness and spatial dynamics.
  • Approximating optimal controls for high-dimensional SRDEs presents significant computational challenges.
  • Existing methods often struggle with the curse of dimensionality and the complexity of feedback control.

Purpose of the Study:

  • To develop a novel numerical algorithm for approximating optimal controls of SRDEs with additive noise.
  • To reduce the computational complexity associated with finding asymptotically optimal control strategies.
  • To provide a framework for analyzing approximation error rates based on structural assumptions.

Main Methods:

  • The algorithm first reformulates the optimal control problem into a feedback control problem.
  • It then approximates the feedback control function using finitely based approximations.
  • Numerical experiments employ artificial neural networks (ANNs) and radial basis function networks (RBFNs) to demonstrate the algorithm's efficacy.

Main Results:

  • The proposed algorithm significantly reduces the computational complexity for finding controls with asymptotically optimal cost.
  • Structural assumptions on approximations yield quantifiable rates for the approximation error of the cost function.
  • Numerical experiments confirm the algorithm's performance and efficiency.

Conclusions:

  • The developed numerical algorithm offers an efficient approach to approximating optimal controls for SRDEs.
  • The method's flexibility allows for application to a broader range of stochastic control problems, including high-dimensional stochastic differential equations.
  • This work advances the field of computational control theory for stochastic systems.