865 research outputs found
Finding unstable periodic orbits: A hybrid approach with polynomial optimization
We present a novel method to compute unstable periodic orbits (UPOs) that optimize the infinite-time average of a given quantity for polynomial ODE systems. The UPO search procedure relies on polynomial optimization to construct nonnegative polynomials whose sublevel sets approximately localize parts of the optimal UPO, and that can be used to implement a simple yet effective control strategy to reduce the UPO's instability. Precisely, we construct a family of controlled ODE systems, parameterized by a scalar k, such that the original ODE system is recovered for k=0 and such that the optimal orbit is less unstable, or even stabilized, for k>0. Periodic orbits for the controlled system can often be more easily converged with traditional methods, and numerical continuation in k allows one to recover optimal UPOs for the original system. The effectiveness of this approach is illustrated on three low-dimensional ODE systems with chaotic dynamics
Non-linear State-space Model Identification from Video Data using Deep Encoders
Identifying systems with high-dimensional inputs and outputs, such as systems measured by video streams, is a challenging problem with numerous applications in robotics, autonomous vehicles and medical imaging. In this paper, we propose a novel non-linear state-space identification method starting from high-dimensional input and output data. Multiple computational and conceptual advances are combined to handle the high-dimensional nature of the data. An encoder function, represented by a neural network, is introduced to learn a reconstructability map to estimate the model states from past inputs and outputs. This encoder function is jointly learned with the dynamics. Furthermore, multiple computational improvements, such as an improved reformulation of multiple shooting and batch optimization, are proposed to keep the computational time under control when dealing with high-dimensional and large datasets. We apply the proposed method to a video stream of a simulated environment of a controllable ball in a unit box. The simulation study shows low simulation error with excellent long term prediction for the obtained model using the proposed method
Towards scalable parallel-in-time turbulent flow simulations
We present a reformulation of unsteady turbulent flow simulations. The initial condition is relaxed and information is allowed to propagate both forward and backward in time. Simulations of chaotic dynamical systems with this reformulation can be proven to be well-conditioned time domain boundary value problems. The reformulation can enable scalable parallel-in-time simulation of turbulent flows.United States. Air Force Office of Scientific Research. Small Business Technology Transfer Program (Contract FA9550-12-C-0065
Memory Clustering Using Persistent Homology for Multimodality- and Discontinuity-Sensitive Learning of Optimal Control Warm-Starts
Shooting methods are an efficient approach to solving nonlinear optimal
control problems. As they use local optimization, they exhibit favorable
convergence when initialized with a good warm-start but may not converge at all
if provided with a poor initial guess. Recent work has focused on providing an
initial guess from a learned model trained on samples generated during an
offline exploration of the problem space. However, in practice the solutions
contain discontinuities introduced by system dynamics or the environment.
Additionally, in many cases multiple equally suitable, i.e., multi-modal,
solutions exist to solve a problem. Classic learning approaches smooth across
the boundary of these discontinuities and thus generalize poorly. In this work,
we apply tools from algebraic topology to extract information on the underlying
structure of the solution space. In particular, we introduce a method based on
persistent homology to automatically cluster the dataset of precomputed
solutions to obtain different candidate initial guesses. We then train a
Mixture-of-Experts within each cluster to predict state and control
trajectories to warm-start the optimal control solver and provide a comparison
with modality-agnostic learning. We demonstrate our method on a cart-pole toy
problem and a quadrotor avoiding obstacles, and show that clustering samples
based on inherent structure improves the warm-start quality.Comment: 12 pages, 10 figures, accepted as a regular paper in IEEE
Transactions on Robotics (T-RO). Supplementary video:
https://youtu.be/lUULTWCFxY8 Code:
https://github.com/wxmerkt/topological_memory_clustering The first two
authors contributed equall
Multi-resolution Tensor Learning for Large-Scale Spatial Data
High-dimensional tensor models are notoriously computationally expensive to
train. We present a meta-learning algorithm, MMT, that can significantly speed
up the process for spatial tensor models. MMT leverages the property that
spatial data can be viewed at multiple resolutions, which are related by
coarsening and finegraining from one resolution to another. Using this
property, MMT learns a tensor model by starting from a coarse resolution and
iteratively increasing the model complexity. In order to not "over-train" on
coarse resolution models, we investigate an information-theoretic fine-graining
criterion to decide when to transition into higher-resolution models. We
provide both theoretical and empirical evidence for the advantages of this
approach. When applied to two real-world large-scale spatial datasets for
basketball player and animal behavior modeling, our approach demonstrate 3 key
benefits: 1) it efficiently captures higher-order interactions (i.e., tensor
latent factors), 2) it is orders of magnitude faster than fixed resolution
learning and scales to very fine-grained spatial resolutions, and 3) it
reliably yields accurate and interpretable models
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