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Results for “Hamilton-Jacobi-Bellman equation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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4 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Whitepaper: Optimal Control from a Fluid Dynamics Perspective

An optimal control problem described by the Hamilton-Jacobi-Bellman equation can be developed into a problem that can be solved by general computational fluid dynamics packages. We describe how this formulation would allow a classical problem in optimal control, Zermelo’s problem, to be treated as a multi-fluid problem. This approach has the advantage of allowing optimal navigation problems to be conducted over large areas, as well as to include moderately larger numbers of ships. We draw comparisons between this approach and the field of fluid control for fluid animations in movies.

42 ENGINEERING↗
DOE OSTI2025

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker–Planck Equations

The Fokker-Planck (FP) equation is a foundational partial differential equation (PDE) in stochastic processes involving Brownian motions. However, the curse of dimensionality (CoD) poses a formidable challenge when dealing with high-dimensional FP equations. Although Monte Carlo simulation and (vanilla) Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit significant numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimensionality increases, surpassing the precision of numerical simulations and resulting in substantial errors. In addition, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a notoriously difficult Hamilton-Jacobi-Bellman (HJB) equation, which is impractical for PINN learning, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in stochastic differential equations (SDEs). The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling, offering an effective means to overcome the CoD. Three fitting methods, Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN, are introduced, each contributing unique advantages in computational complexity, accuracy, and generality. The proposed score-based SDE solver operates in two stages: first, employing score matching or Score-PINN to acquire the score function; and second, solving the LL via an ordinary differential equation (ODE) using the obtained score function. Comparative evaluations across these methods showcase varying trade-offs. The proposed methodology is evaluated across diverse SDEs, including anisotropic Ornstein-Uhlenbeck processes, geometric Brownian motion, and Brownian motion with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy distributions. The numerical results demonstrate the score-based SDE solver’s stability, speed, and performance across different experimental settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Neural network approaches for parameterized optimal control

Here, we consider numerical approaches for deterministic, finite-dimensional optimal control problems whose dynamics depend on unknown or uncertain parameters. We seek to amortize the solution over a set of relevant parameters in an offline stage to enable rapid decision-making and be able to react to changes in the parameter in the online stage. To tackle the curse of dimensionality arising when the state and/or parameter are high-dimensional, we represent the policy using neural networks. We compare two training paradigms: First, our model-based approach leverages the dynamics and definition of the objective function to learn the value function of the parameterized optimal control problem and obtain the policy using a feedback form. Second, we use actor-critic reinforcement learning to approximate the policy in a data-driven way. Using an example involving a two-dimensional convection-diffusion equation, which features high-dimensional state and parameter spaces, we investigate the accuracy and efficiency of both training paradigms. While both paradigms lead to a reasonable approximation of the policy, the model-based approach is more accurate and considerably reduces the number of PDE solves.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Bias-Variance Trade-Off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.

97 MATHEMATICS AND COMPUTING↗