MINs can scale to high-dimensional input spaces and leverage offline logged data for both contextual and non-contextual optimization problems.

Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys methodological and theoretical developments at the intersection of operator learning and inverse problems. It begins by summarizing the probabilistic and deterministic approaches to inverse problems, and pays special attention to emerging measure-centric formulations that treat observed data or unknown parameters as probability distributions. The discussion then turns to operator learning by covering essential components such as data generation, loss functions, and widely used architectures for representing function-to-function maps. The core of the chapter centers on the end-to-end inverse operator learning paradigm, which aims to directly map observed data to the solution of the inverse problem without requiring explicit knowledge of the forward map. It highlights the unique challenge that noise plays in this data-driven inversion setting, presents structure-aware architectures for both point predictions and posterior estimates, and surveys relevant theory for linear and nonlinear inverse problems. The chapter also discusses the estimation of priors and regularizers, where operator learning is used more selectively within classical inversion algorithms.

MINs can scale to high-dimensional input spaces and leverage offline logged data for both contextual and non-contextual optimization problems.

Knowledge of a manipulator's workspace is fundamental for a variety of tasks including robot design, grasp planning and robot base placement. Consequently, workspace representations are well studied in robotics. Two important representations are reachability maps and inverse reachability maps. The former predicts whether a given end-effector pose is reachable from where the robot currently is, and the latter suggests suitable base positions for a desired end-effector pose. Typically, the reachability map is built by discretizing the 6D space containing the robot's workspace and determining, for each cell, whether it is reachable or not. The reachability map is subsequently inverted to build the inverse map. This is a cumbersome process which restricts the applications of such maps. In this work, we exploit commonalities of existing six and seven axis robot arms to reduce the dimension of the discretization from 6D to 4D. We propose Reachability Map 4D (RM4D), a map that only requires a single 4D data structure for both forward and inverse queries. This gives a much more compact map that can be constructed by an order of magnitude faster than existing maps, with no inversion overheads and no loss in accuracy. Our experiments showcase the usefulness of RM4D for grasp planning with a mobile manipulator.

Inverse problems, i.e., estimating parameters of physical models from experimental data, are ubiquitous in science and engineering. The Bayesian formulation is the gold standard because it alleviates ill-posedness issues and quantifies epistemic uncertainty. Since analytical posteriors are not typically available, one resorts to Markov chain Monte Carlo sampling or approximate variational inference. However, inference needs to be rerun from scratch for each new set of data. This drawback limits the applicability of the Bayesian formulation to real-time settings, e.g., health monitoring of engineered systems, and medical diagnosis. The objective of this paper is to develop a methodology that enables real-time inference by learning the Bayesian inverse map, i.e., the map from data to posteriors. Our approach is as follows. We represent the posterior distribution using a parameterization based on deep neural networks. Next, we learn the network parameters by amortized variational inference method which involves maximizing the expectation of evidence lower bound over all possible datasets compatible with the model. We demonstrate our approach by solving examples a set of benchmark problems from science and engineering. Our results show that the posterior estimates of our approach are in agreement with the corresponding ground truth obtained by Markov chain Monte Carlo. Once trained, our approach provides the posterior parameters of observation just at the cost of a forward pass of the neural network.

Deep Learning (DL), in particular deep neural networks (DNN), by design is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties--such as stability, conservation, and positivity--and desired accuracy need to be achieved. DL methods in their original forms are not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics and hence obtaining higher accuracy. This short communication introduces several model-constrained DL approaches--including both feed-forward DNN and autoencoders--that are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems. We present and provide intuitions for our formulations for general nonlinear problems. For linear inverse problems and linear networks, the first order optimality conditions show that our model-constrained DL approaches can learn information encoded in the underlying mathematical models, and thus can produce consistent or equivalent inverse solutions, while naive purely data-based counterparts cannot.

In this work, we aim to solve data-driven optimization problems, where the goal is to find an input that maximizes an unknown score function given access to a dataset of inputs with corresponding scores. When the inputs are high-dimensional and valid inputs constitute a small subset of this space (e.g., valid protein sequences or valid natural images), such model-based optimization problems become exceptionally difficult, since the optimizer must avoid out-of-distribution and invalid inputs. We propose to address such problem with model inversion networks (MINs), which learn an inverse mapping from scores to inputs. MINs can scale to high-dimensional input spaces and leverage offline logged data for both contextual and non-contextual optimization problems. MINs can also handle both purely offline data sources and active data collection. We evaluate MINs on tasks from the Bayesian optimization literature, high-dimensional model-based optimization problems over images and protein designs, and contextual bandit optimization from logged data.

Have you ever wondered how human-like a robot can become? Researchers are one step closer, literally, to machines having more human-like capabilities. A cross-disciplinary research team from the University of Southern California (USC) departments of engineering (biomedical, electrical, aerospace and mechanical), computer science, biokinesiology, and physical therapy joined forces to create a robot that can teach itself to walk. Valero-Cuevas published their findings recently in Nature Machine Intelligence on March 11, 2019. The researchers created a "biologically plausible algorithm" called "G2P" (general to particular).