DeepReach: A Deep Learning Approach to High-Dimensional Reachability

• URL: http://arxiv.org/abs/2011.02082v1
• Date: Wed, 4 Nov 2020 00:47:59 GMT
• Title: DeepReach: A Deep Learning Approach to High-Dimensional Reachability
• Authors: Somil Bansal, Claire Tomlin
• Abstract summary: Hamilton-Jacobi (HJ) reachability analysis is an important formal verification method for guaranteeing performance and safety properties of dynamical control systems. We propose DeepReach, a neural PDE solver for high-dimensional reachability problems.
• Score: 6.604421202391151
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Hamilton-Jacobi (HJ) reachability analysis is an important formal verification method for guaranteeing performance and safety properties of dynamical control systems. Its advantages include compatibility with general nonlinear system dynamics, formal treatment of bounded disturbances, and the ability to deal with state and input constraints. However, it involves solving a PDE, whose computational and memory complexity scales exponentially with respect to the number of state variables, limiting its direct use to small-scale systems. We propose DeepReach, a method that leverages new developments in sinusoidal networks to develop a neural PDE solver for high-dimensional reachability problems. The computational requirements of DeepReach do not scale directly with the state dimension, but rather with the complexity of the underlying reachable tube. DeepReach achieves comparable results to the state-of-the-art reachability methods, does not require any explicit supervision for the PDE solution, can easily handle external disturbances, adversarial inputs, and system constraints, and also provides a safety controller for the system. We demonstrate DeepReach on a 9D multi-vehicle collision problem, and a 10D narrow passage problem, motivated by autonomous driving applications.

Related papers

• Interpretable and Efficient Data-driven Discovery and Control of Distributed Systems [1.5195865840919498]
  Reinforcement Learning (RL) has emerged as a promising control paradigm for systems with high-dimensional, nonlinear dynamics. We propose a data-efficient, interpretable, and scalable framework for PDE control.
  arXiv  Detail & Related papers  (2024-11-06T18:26:19Z)
• Deep Autoencoder with SVD-Like Convergence and Flat Minima [1.0742675209112622]
  We propose a learnable weighted hybrid autoencoder to overcome the Kolmogorov barrier. We empirically find that our trained model has a sharpness thousands of times smaller compared to other models.
  arXiv  Detail & Related papers  (2024-10-23T00:04:26Z)
• Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning [0.0]
  In the absence of labeled datasets, we utilize the PDE residual loss to learn the physical system, an approach known as physics-informed DeepONet. This method faces significant computational challenges, primarily due to the curse of dimensionality, as the computational cost increases exponentially with finer discretization. We introduce the Separable DeepONet framework to address these challenges and improve scalability for high-dimensional PDEs.
  arXiv  Detail & Related papers  (2024-07-21T16:33:56Z)
• Scaling #DNN-Verification Tools with Efficient Bound Propagation and Parallel Computing [57.49021927832259]
  Deep Neural Networks (DNNs) are powerful tools that have shown extraordinary results in many scenarios. However, their intricate designs and lack of transparency raise safety concerns when applied in real-world applications. Formal Verification (FV) of DNNs has emerged as a valuable solution to provide provable guarantees on the safety aspect.
  arXiv  Detail & Related papers  (2023-12-10T13:51:25Z)
• The Hard-Constraint PINNs for Interface Optimal Control Problems [1.6237916898865998]
  We show that PINNs can be applied to solve optimal control problems with interfaces and some control constraints. The resulting algorithm is mesh-free and scalable to different PDEs.
  arXiv  Detail & Related papers  (2023-08-13T07:56:01Z)
• A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
  We develop an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters. We show that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors.
  arXiv  Detail & Related papers  (2023-04-28T17:28:18Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Generating Formal Safety Assurances for High-Dimensional Reachability [4.523089386111081]
  Hamilton-Jacobi (HJ) reachability analysis is a popular formal verification tool for providing safety and performance guarantees for autonomous systems. A recently proposed method called DeepReach overcomes this challenge by leveraging a sinusoidal neural PDE solver for high-dimensional reachability problems. We propose a method to compute an error bound for the DeepReach solution, resulting in a safe approximation of the true reachable tube.
  arXiv  Detail & Related papers  (2022-09-25T22:15:53Z)
• Chance-Constrained Control with Lexicographic Deep Reinforcement Learning [77.34726150561087]
  This paper proposes a lexicographic Deep Reinforcement Learning (DeepRL)-based approach to chance-constrained Markov Decision Processes. A lexicographic version of the well-known DeepRL algorithm DQN is also proposed and validated via simulations.
  arXiv  Detail & Related papers  (2020-10-19T13:09:14Z)
• Learning to Control PDEs with Differentiable Physics [102.36050646250871]
  We present a novel hierarchical predictor-corrector scheme which enables neural networks to learn to understand and control complex nonlinear physical systems over long time frames. We demonstrate that our method successfully develops an understanding of complex physical systems and learns to control them for tasks involving PDEs.
  arXiv  Detail & Related papers  (2020-01-21T11:58:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.