Optimal trajectory calculation using neural networks
Bibliographic record
Abstract
Optimal control methods for linear systems have reached a substantial level of maturity, both in terms of conceptual understanding and scalable computational implementation. For non-linear systems, an open-loop feedback control may be calculated using Pontryagin's Maximum Principle. Alternatively, the Hamilton-Jacobi-Bellman (HJB) equation may be used to calculate the optimal control in a state-feedback form. However, it is an established fact that this equation becomes progressively harder to solve as the number of state variables increases. In this thesis, we discuss a Neural Network (NN)-based method [1] to approximate the solution to the HJB equation arising from high-dimensional ODE systems. We leverage the equivalency between the HJB equation and Pontryagin's Principle to generate the training and test datasets and define a physics-based loss function. The NN is then trained using a supervised optimization approach. We also examine an existing toolkit [2] to approximate the optimal control based on a power series expansion of the system around an equilibrium point in an infinite time horizon setting. We examine the possibility of incorporating this toolkit in the NN training procedure at different stages. The proposed methods are applied to three problems: optimal control of a 6 degree-of-freedom rigid body and the stabilization of ODE systems arising from the discretization of a Burgers'-like non-linear PDE and the damped wave equation. \n \nReferences: \n[1] Tenavi Nakamura-Zimmerer, Qi Gong, and Wei Kang. Adaptive deep learning for high-dimensional Hamilton-Jacobi-Bellman equations. SIAM Journal of Scientific Computing, 43(2):A1221–A1247, 2021. \n[2] Arthur J. Krener. Nonlinear systems toolbox v.1.0, 1997. MATLAB based toolbox available by request from ajkrener@ucdavis.edu
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".