An official website of the United States government
Finding Nash equilibrial policies for two-player differential games requires solving Hamilton-Jacobi-Isaacs (HJI) PDEs. Self-supervised learning has been used to approximate solutions of such PDEs while circumventing the curse of dimensionality. However, this method fails to learn discontinuous PDE solutions due to its sampling nature, leading to poor safety performance of the resulting controllers in robotics applications when player rewards are discontinuous. This paper investigates two potential solutions to this problem: a hybrid method that leverages both supervised Nash equilibria and the HJI PDE, and a value-hardening method where a sequence of HJIs are solved with a gradually hardening reward. We compare these solutions using the resulting generalization and safety performance in two vehicle interaction simulation studies with 5D and 9D state spaces, respectively. Results show that with informative supervision (e.g., collision and near-collision demonstrations) and the low cost of self-supervised learning, the hybrid method achieves better safety performance than the supervised, self-supervised, and value hardening approaches on equal computational budget. Value hardening fails to generalize in the higher-dimensional case without informative supervision. Lastly, we show that the neural activation function needs to be continuously differentiable for learning PDEs and its choice can be case dependent.
more »
« less
This content will become publicly available on January 28, 2027
On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights
We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its $$\overline{\partial}$$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative.
more »
« less
- Award ID(s):
- 2306438
- PAR ID:
- 10692789
- Publisher / Repository:
- SIGMA
- Date Published:
- Journal Name:
- Symmetry, Integrability and Geometry: Methods and Applications
- ISSN:
- 1815-0659
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A new way of encoding a non-self-adjoint Jacobi matrix J by a spectral measure of |J| together with a phase function was described by Pushnitski–Štampach in the bounded case. We present another perspective on this correspondence, based on Weyl functions instead of moments, which simplifies some proofs and generalizes the correspondence to the unbounded case. In particular, we find a bijection between proper Jacobi matrices with positive off-diagonal elements, and a class of spectral data. We prove that this mapping is continuous in a suitable sense. To prove injectivity of the map, we prove a local Borg–Marchenko theorem for unbounded non-self-adjoint Jacobi matrices in this class that may be of independent interest.more » « less
-
On a smooth, compact, Riemannian manifold without boundary(M,g), let\Delta_{g}be the Laplace–Beltrami operator. We define the orthogonal projection operator \Pi_{I_\lambda}\colon L^{2}(M)\to \bigoplus_{\mathclap{\lambda_j\in I_\lambda}}\ker(\Delta_{g}+\lambda_{j}^{2}) for an intervalI_{\lambda}centered around\lambda\in\Rof a small, fixed length. The Schwartz kernel,\Pi_{I_\lambda}(x,y), of this operator plays a key role in the analysis of monochromatic random waves, a model for high energy eigenfunctions. It is expected that\Pi_{I_\lambda}(x,y)has universal asymptotics as\lambda \to \inftyin a shrinking neighborhood of the diagonal inM\times M(providedI_{\lambda}is chosen appropriately) and hence that certain statistics for monochromatic random waves have universal behavior. These asymptotics are well known for the torus and the round sphere, and were recently proved to hold near points inMwith few geodesic loops by Canzani–Hanin. In this article, we prove that the same universal asymptotics hold in the opposite case of Zoll manifolds (manifolds all of whose geodesics are closed with a common period) under an assumption on the volume of loops with length incommensurable with the minimal common period.more » « less
-
null (Ed.)Abstract We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm–Liouville eigentransforms and calculating Gauss–Jacobi quadrature rules. Our approach, which applies in the case in which both of the parameters $$\alpha $$ and $$\beta $$ in Jacobi’s differential equation are of magnitude less than $1/2$, is based on the well-known fact that in this regime Jacobi’s differential equation admits a nonoscillatory phase function that can be loosely approximated via an affine function over much of its domain. We illustrate this with several numerical experiments, the source code for which is publicly available.more » « less
-
We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semiexplicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph’s normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, that is, when the degrees diverge in a suitable sense. Although equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.more » « less
