Recent years have witnessed the promise of coupling machine learning methods and physical domain-specific insight for solving scientific problems based on partial differential equations (PDEs). However, being data-intensive, these methods still require a large amount of PDE data. This reintroduces the need for expensive numerical PDE solutions, partially undermining the original goal of avoiding these expensive simulations. In this work, seeking data efficiency, we design unsupervised pretraining and in-context learning methods for PDE operator learning. To reduce the need for training data with simulated solutions, we pretrain neural operators on unlabeled PDE data using reconstruction-based proxy tasks. To improve out-of-distribution performance, we further assist neural operators in flexibly leveraging in-context learning methods, without incurring extra training costs or designs. Extensive empirical evaluations on a diverse set of PDEs demonstrate that our method is highly data-efficient, more generalizable, and even outperforms conventional vision-pretrained models.
In this paper, we present a method of multi-robot motion planning by biasing centralized, sampling-based tree search with decentralized, data-driven steer and distance heuristics. Over a range of robot and obstacle densities, we evaluate the plain Rapidly-expanding Random Trees (RRT), and variants of our method for double integrator dynamics. We show that whereas plain RRT fails in every instance to plan for $4$ robots, our method can plan for up to 16 robots, corresponding to searching through a very large 65-dimensional space, which validates the effectiveness of data-driven heuristics at combating exponential search space growth. We also find that the heuristic information is complementary; using both heuristics produces search trees with lower failure rates, nodes, and path costs when compared to using each in isolation. These results illustrate the effective decomposition of high-dimensional joint-space motion planning problems into local problems.
We study Bayesian optimization (BO) in high-dimensional and non-stationary scenarios. Existing algorithms for such scenarios typically require extensive hyperparameter tuning, which limits their practical effectiveness. We propose a framework, called BALLET, which adaptively filters for a high-confidence region of interest (ROI) as a superlevel-set of a nonparametric probabilistic model such as a Gaussian process (GP). Our approach is easy to tune, and is able to focus on local region of the optimization space that can be tackled by existing BO methods. The key idea is to use two probabilistic models: a coarse GP to identify the ROI, and a localized GP for optimization within the ROI. We show theoretically that BALLET can efficiently shrink the search space, and can exhibit a tighter regret bound than standard BO without ROI filtering. We demonstrate empirically the effectiveness of BALLET on both synthetic and real-world optimization tasks.
We present MLNav, a learning-enhanced path planning framework for safety-critical and resource-limited systems operating in complex environments, such as rovers navigating on Mars. MLNav makes judicious use of machine learning to enhance the efficiency of path planning while fully respecting safety constraints. In particular, the dominant computational cost in such safety-critical settings is running a model-based safety checker on the proposed paths. Our learned search heuristic can simultaneously predict the feasibility for all path options in a single run, and the model-based safety checker is only invoked on the top-scoring paths. We validate in high-fidelity simulations using both real Martian terrain data collected by the Perseverance rover, as well as a suite of challenging synthetic terrains. Our experiments show that: (i) compared to the baseline ENav path planner on board the Perserverance rover, MLNav can provide a significant improvement in multiple key metrics, such as a 10x reduction in collision checks when navigating real Martian terrains, despite being trained with synthetic terrains; and (ii) MLNav can successfully navigate highly challenging terrains where the baseline ENav fails to find a feasible path before timing out.
We propose a machine learning approach for quickly solving Mixed Integer Programs (MIP) by learning to prioritize a set of decision variables, which we call pseudo-backdoors, for branching that results in faster solution times. Learning-based approaches have seen success in the area of solving combinatorial optimization problems by being able to flexibly leverage common structures in a given distribution of problems. Our approach takes inspiration from the concept of strong backdoors, which corresponds to a small set of variables such that only branching on these variables yields an optimal integral solution and a proof of optimality. Our notion of pseudo-backdoors corresponds to a small set of variables such that only branching on them leads to faster solve time (which can be solver dependent). A key advantage of pseudo-backdoors over strong backdoors is that they are much amenable to data-driven identification or prediction. Our proposed method learns to estimate the solver performance of a proposed pseudo-backdoor, using a labeled dataset collected on a set of training MIP instances. This model can then be used to identify high-quality pseudo-backdoors on new MIP instances from the same distribution. We evaluate our method on the generalized independent set problems and find that our approach can efficiently identify high-quality pseudo-backdoors. In addition, we compare our learned approach against Gurobi, a state-of-the-art MIP solver, demonstrating that our method can be used to improve solver performance.
Enhanced AutoNav (ENav), the baseline surface navigation software for NASA's Perseverance rover, sorts a list of candidate paths for the rover to traverse, then uses the Approximate Clearance Evaluation (ACE) algorithm to evaluate whether the most highly ranked paths are safe. ACE is crucial for maintaining the safety of the rover, but is computationally expensive. If the most promising candidates in the list of paths are all found to be infeasible, ENav must continue to search the list and run time-consuming ACE evaluations until a feasible path is found. In this paper, we present two heuristics that, given a terrain heightmap around the rover, produce cost estimates that more effectively rank the candidate paths before ACE evaluation. The first heuristic uses Sobel operators and convolution to incorporate the cost of traversing high-gradient terrain. The second heuristic uses a machine learning (ML) model to predict areas that will be deemed untraversable by ACE. We used physics simulations to collect training data for the ML model and to run Monte Carlo trials to quantify navigation performance across a variety of terrains with various slopes and rock distributions. Compared to ENav's baseline performance, integrating the heuristics can lead to a significant reduction in ACE evaluations and average computation time per planning cycle, increase path efficiency, and maintain or improve the rate of successful traverses. This strategy of targeting specific bottlenecks with ML while maintaining the original ACE safety checks provides an example of how ML can be infused into planetary science missions and other safety-critical software.
This paper studies how to design abstractions of large-scale combinatorial optimization problems that can leverage existing state-of-the-art solvers in general purpose ways, and that are amenable to data-driven design. The goal is to arrive at new approaches that can reliably outperform existing solvers in wall-clock time. We focus on solving integer programs, and ground our approach in the large neighborhood search (LNS) paradigm, which iteratively chooses a subset of variables to optimize while leaving the remainder fixed. The appeal of LNS is that it can easily use any existing solver as a subroutine, and thus can inherit the benefits of carefully engineered heuristic approaches and their software implementations. We also show that one can learn a good neighborhood selector from training data. Through an extensive empirical validation, we demonstrate that our LNS framework can significantly outperform, in wall-clock time, compared to state-of-the-art commercial solvers such as Gurobi.
We study the problem of learning sequential decision-making policies in settings with multiple state-action representations. Such settings naturally arise in many domains, such as planning (e.g., multiple integer programming formulations) and various combinatorial optimization problems (e.g., those with both integer programming and graph-based formulations). Inspired by the classical co-training framework for classification, we study the problem of co-training for policy learning. We present sufficient conditions under which learning from two views can improve upon learning from a single view alone. Motivated by these theoretical insights, we present a meta-algorithm for co-training for sequential decision making. Our framework is compatible with both reinforcement learning and imitation learning. We validate the effectiveness of our approach across a wide range of tasks, including discrete/continuous control and combinatorial optimization.
We apply numerical methods in combination with finite-difference-time-domain (FDTD) simulations to optimize transmission properties of plasmonic mirror color filters using a multi-objective figure of merit over a five-dimensional parameter space by utilizing novel multi-fidelity Gaussian processes approach. We compare these results with conventional derivative-free global search algorithms, such as (single-fidelity) Gaussian Processes optimization scheme, and Particle Swarm Optimization---a commonly used method in nanophotonics community, which is implemented in Lumerical commercial photonics software. We demonstrate the performance of various numerical optimization approaches on several pre-collected real-world datasets and show that by properly trading off expensive information sources with cheap simulations, one can more effectively optimize the transmission properties with a fixed budget.
How can we efficiently gather information to optimize an unknown function, when presented with multiple, mutually dependent information sources with different costs? For example, when optimizing a robotic system, intelligently trading off computer simulations and real robot testings can lead to significant savings. Existing methods, such as multi-fidelity GP-UCB or Entropy Search-based approaches, either make simplistic assumptions on the interaction among different fidelities or use simple heuristics that lack theoretical guarantees. In this paper, we study multi-fidelity Bayesian optimization with complex structural dependencies among multiple outputs, and propose MF-MI-Greedy, a principled algorithmic framework for addressing this problem. In particular, we model different fidelities using additive Gaussian processes based on shared latent structures with the target function. Then we use cost-sensitive mutual information gain for efficient Bayesian global optimization. We propose a simple notion of regret which incorporates the cost of different fidelities, and prove that MF-MI-Greedy achieves low regret. We demonstrate the strong empirical performance of our algorithm on both synthetic and real-world datasets.