Learning with identical train and test distributions has been extensively investigated both practically and theoretically. Much remains to be understood, however, in statistical learning under distribution shifts. This paper focuses on a distribution shift setting where train and test distributions can be related by classes of (data) transformation maps. We initiate a theoretical study for this framework, investigating learning scenarios where the target class of transformations is either known or unknown. We establish learning rules and algorithmic reductions to Empirical Risk Minimization (ERM), accompanied with learning guarantees. We obtain upper bounds on the sample complexity in terms of the VC dimension of the class composing predictors with transformations, which we show in many cases is not much larger than the VC dimension of the class of predictors. We highlight that the learning rules we derive offer a game-theoretic viewpoint on distribution shift: a learner searching for predictors and an adversary searching for transformation maps to respectively minimize and maximize the worst-case loss.

Learning with identical train and test distributions has been extensively investigated both practically and theoretically. Much remains to be understood, however, in statistical learning under distribution shifts. This paper focuses on a distribution shift setting where train and test distributions can be related by classes of (data) transformation maps. We initiate a theoretical study for this framework, investigating learning scenarios where the target class of transformations is either known or unknown. We establish learning rules and algorithmic reductions to Empirical Risk Minimization (ERM), accompanied with learning guarantees.

This paper proposes a novel learning approach for designing Kazantzis-Kravaris/Luenberger (KKL) observers for autonomous nonlinear systems. The design of a KKL observer involves finding an injective map that transforms the system state into a higher-dimensional observer state, whose dynamics is linear and stable. The observer's state is then mapped back to the original system coordinates via the inverse map to obtain the state estimate. However, finding this transformation and its inverse is quite challenging. We propose to sequentially approximate these maps by neural networks that are trained using physics-informed learning. We generate synthetic data for training by numerically solving the system and observer dynamics. Theoretical guarantees for the robustness of state estimation against approximation error and system uncertainties are provided. Additionally, a systematic method for optimizing observer performance through parameter selection is presented. The effectiveness of the proposed approach is demonstrated through numerical simulations on benchmark examples and its application to sensor fault detection and isolation in a network of Kuramoto oscillators using learned KKL observers.

In modern feedback control systems theory and practice, reliable access to the dynamically evolving system states is needed at both the implementation stage of advanced control algorithms and for process/system condition and performance monitoring purposes [16, 8, 39, 13, 47]. Traditionally, an explicit use of an available dynamic model complemented by sensor measurements, involving measurable physical and chemical variables of the system of interest, represented a first option to respond to the above need. However, in practice, key critical state variables are often not available for direct on-line measurement, due to inherent physical as well as practically insurmountable technical and economic limitations associated with the current state of sensor technology as it is invariably deployed in cases of considerable system complexity [47, 16, 8, 39]. In light of the above remarks, a better, scientifically sound and practically insightful option is the design of a state estimator (an observer). This is itself an appropriately structured dynamical system itself that utilizes all information provided by a system model as well as available sensor measurements to accurately reconstruct the dynamic profiles of all other unmeasurable state variables [47, 16, 8, 13].

Designing Luenberger observers for nonlinear systems involves the challenging task of transforming the state to an alternate coordinate system, possibly of higher dimensions, where the system is asymptotically stable and linear up to output injection. The observer then estimates the system's state in the original coordinates by inverting the transformation map. However, finding a suitable injective transformation whose inverse can be derived remains a primary challenge for general nonlinear systems. We propose a novel approach that uses supervised physics-informed neural networks to approximate both the transformation and its inverse. Our method exhibits superior generalization capabilities to contemporary methods and demonstrates robustness to both neural network's approximation errors and system uncertainties.

We analyze the oracle complexity of sampling from polynomially decaying heavy-tailed target densities based on running the Unadjusted Langevin Algorithm on certain transformed versions of the target density. The specific class of closed-form transformation maps that we construct are shown to be diffeomorphisms, and are particularly suited for developing efficient diffusion-based samplers. We characterize the precise class of heavy-tailed densities for which polynomial-order oracle complexities (in dimension and inverse target accuracy) could be obtained, and provide illustrative examples. We highlight the relationship between our assumptions and functional inequalities (super and weak Poincar\'e inequalities) based on non-local Dirichlet forms defined via fractional Laplacian operators, used to characterize the heavy-tailed equilibrium densities of certain stable-driven stochastic differential equations.

In 2016, World Economic Forum (WEF) founder and Executive Chairman Klaus Schwab proclaimed the fourth industrial revolution as a distinct evolution from its predecessor because of the rapid onset of ubiquitous change. This revolution -- the current environment in which disruptive technologies such as artificial intelligence, cloud computing and the "internet of things," among others -- is profoundly changing the way we live and work. The complexity and scale of such change have seen the need for new means and approaches to linking intelligence, understanding and specialists at the global level. Transformation Maps, a collaborative digital tool developed by the WEF available in English, Mandarin, Spanish, Arabic and Japanese that harnesses knowledge, charts interactions and analyzes links between industries, countries and issues that are shaping the world, may very well be the platform to do so. According to Jeremy Jurgens, managing director and head of knowledge and digital engagement at the Swiss-based non-profit, Transformation Maps are certainly a solution made only available because of the accelerated change brought about as part of the current industrial revolution.