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We present a neural network approach for approximating the value function of high- dimensional stochastic control problems. Our training process simultaneously updates our value function estimate and identifies the part of the state space likely to be visited by optimal trajectories. Our approach leverages insights from optimal control theory and the fundamental relation between semi-linear parabolic partial differential equations and forward-backward stochastic differential equations. To focus the sampling on relevant states during neural network training, we use the stochastic Pontryagin maximum principle (PMP) to obtain the optimal controls for the current value function estimate. By design, our approach coincides with the method of characteristics for the non-viscous Hamilton-Jacobi-Bellman equation arising in deterministic control problems. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the HJB equations along the sampled trajectories. Importantly, training is unsupervised in that it does not require solutions of the control problem. Our numerical experiments highlight our scheme’s ability to identify the relevant parts of the state space and produce meaningful value estimates. Using a two-dimensional model problem, we demonstrate the importance of the stochastic PMP to inform the sampling and compare to a finite element approach. With a nonlinear control affine quadcopter example, we illustrate that our approach can handle complicated dynamics. For a 100-dimensional benchmark problem, we demonstrate that our approach improves accuracy and time-to-solution and, via a modification, we show the wider applicability of our scheme.
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This content will become publicly available on January 1, 2026
Deep neural network for solving Poisson-Boltzmann equations on protein surfaces
In this paper, we propose a new deep learning method for the nonlinear Poisson-Boltzmann problems with applications in computational biology. To tackle the discontinuity of the solution, e.g., across protein surfaces, we approximate the solution by a piecewise mesh-free neural network that can capture the dramatic change in the solution across the interface. The partial differential equation problem is first reformulated as a least-squares physics-informed neural network (PINN)-type problem and then discretized to an objective function using mean squared error via sampling. The solution is obtained by minimizing the designed objective function via standard training algorithms such as the stochastic gradient descent method. Finally, the effectiveness and efficiency of the neural network are validated using complex protein interfaces on various manufactured functions with different frequencies.
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- Award ID(s):
- 2309557
- PAR ID:
- 10626626
- Publisher / Repository:
- Begell House
- Date Published:
- Journal Name:
- Journal of Machine Learning for Modeling and Computing
- Volume:
- 6
- Issue:
- 1
- ISSN:
- 2689-3967
- Page Range / eLocation ID:
- 41 to 63
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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