An official website of the United States government
This paper presents a method that uses neural networks as a caching mechanism to reduce the variance of Monte Carlo Partial Differential Equation solvers, such as the Walk-on-Spheres algorithm [Sawhney and Crane 2020]. While these Monte Carlo PDE solvers have the merits of being unbiased and discretization-free, their high variance often hinders real-time applications. On the other hand, neural networks can approximate the PDE solution, and evaluating these networks at inference time can be very fast. However, neural-network-based solutions may suffer from convergence difficulties and high bias. Our hybrid system aims to combine these two potentially complementary solutions by training a neural field to approximate the PDE solution using supervision from a WoS solver. This neural field is then used as a cache in the WoS solver to reduce variance during inference. We demonstrate that our neural field training procedure is better than the commonly used self-supervised objectives in the literature. We also show that our hybrid solver exhibits lower variance than WoS with the same computational budget: it is significantly better for small compute budgets and provides smaller improvements for larger budgets, reaching the same performance as WoS in the limit.
more »
« less
Approximating Discontinuous Nash Equilibrial Values of Two-Player General-Sum Differential Games
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
- PAR ID:
- 10432750
- Date Published:
- Journal Name:
- 2023 IEEE International Conference on Robotics and Automation (ICRA)
- Page Range / eLocation ID:
- 3022 to 3028
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The values of two-player general-sum differential games are viscosity solutions to Hamilton-Jacobi-Isaacs (HJI) equations. Value and policy approximations for such games suffer from the curse of dimensionality (CoD). Alleviating CoD through physics-informed neural networks (PINN) encounters convergence issues when value discontinuity is present due to state constraints. On top of these challenges, it is often necessary to learn generalizable values and policies across a parametric space of games, eg, for game parameter inference when information is incomplete. To address these challenges, we propose in this paper a Pontryagin-mode neural operator that outperforms existing state-of-the-art (SOTA) on safety performance across games with parametric state constraints. Our key contribution is the introduction of a costate loss defined on the discrepancy between forward and backward costate rollouts, which are computationally cheap. We show that the discontinuity of costate dynamics (in the presence of state constraints) effectively enables the learning of discontinuous values, without requiring manually supervised data as suggested by the current SOTA. More importantly, we show that the close relationship between costates and policies makes the former critical in learning feedback control policies with generalizable safety performance.more » « less
-
Learning problems commonly exhibit an interesting feedback mechanism wherein the population data reacts to competing decision makers’ actions. This paper formulates a new game theoretic framework for this phenomenon, called multi-player performative prediction. We focus on two distinct solution concepts, namely (i) performatively stable equilibria and (ii) Nash equilibria of the game. The latter equilibria are arguably more informative, but are generally computationally difficult to find since they are solutions of nonmonotone games. We show that under mild assumptions, the performatively stable equilibria can be found efficiently by a variety of algorithms, including repeated retraining and the repeated (stochastic) gradient method. We then establish transparent sufficient conditions for strong monotonicity of the game and use them to develop algorithms for finding Nash equilibria. We investigate derivative free methods and adaptive gradient algorithms wherein each player alternates between learning a parametric description of their distribution and gradient steps on the empirical risk. Synthetic and semi-synthetic numerical experiments illustrate the results.more » « less
-
Safety and efficiency are primary goals of air traffic management. With the integration of unmanned aerial vehicles (UAVs) into the airspace, UAV traffic management (UTM) has attracted significant interest in the research community to maintain the capacity of three-dimensional (3D) airspace, provide information, and avoid collisions. We propose a new decision-making architecture for UAVs to avoid collision by formulating the problem into a multi-agent game in a 3D airspace. In the proposed game-theoretic approach, the Ego UAV plays a repeated two-player normal-form game, and the payoff functions are designed to capture both the safety and efficiency of feasible actions. An optimal decision in the form of Nash equilibrium (NE) is obtained. Simulation studies are conducted to demonstrate the performance of the proposed game-theoretic collision avoidance approach in several representative multi-UAV scenarios.more » « less
-
Humans excel at efficiently navigating through crowds without collision by focusing on specific visual regions relevant to navigation. However, most robotic visual navigation methods rely on deep learning models pre-trained on vision tasks, which prioritize salient objects—not necessarily relevant to navigation and potentially misleading. Alternative approaches train specialized navigation models from scratch, requiring significant computation. On the other hand, self-supervised learning has revolutionized computer vision and natural language processing, but its application to robotic navigation remains underexplored due to the difficulty of defining effective self-supervision signals. Motivated by these observations, in this work, we propose a Self-Supervised Vision-Action Model for Visual Navigation Pre-Training (VANP). Instead of detecting salient objects that are beneficial for tasks such as classification or detection, VANP learns to focus only on specific visual regions that are relevant to the navigation task. To achieve this, VANP uses a history of visual observations, future actions, and a goal image for self-supervision, and embeds them using two small Transformer Encoders. Then, VANP maximizes the information between the embeddings by using a mutual information maximization objective function. We demonstrate that most VANP-extracted features match with human navigation intuition. VANP achieves comparable performance as models learned end-to-end with half the training time and models trained on a large-scale, fully supervised dataset, i.e., ImageNet, with only 0.08% data.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ICRA48891.2023.10160219
