Well known to the machine learning community, the random feature model, originally introduced by Rahimi and Recht in 2008, is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to the real line. In this paper, we instead propose a methodology for use of the random feature model as a data-driven surrogate for operators that map an input Banach space to an output Banach space. Although the methodology is quite general, we consider operators defined by partial differential equations (PDEs); here, the inputs and outputs are themselves functions, with the input parameters being functions required to specify the problem, such as initial data or coefficients, and the outputs being solutions of the problem. Upon discretization, the model inherits several desirable attributes from this infinite-dimensional, function space viewpoint, including mesh-invariant approximation error with respect to the true PDE solution map and the capability to be trained at one mesh resolution and then deployed at different mesh resolutions. We view the random feature model as a non-intrusive data-driven emulator, provide a mathematical framework for its interpretation, and demonstrate its ability to efficiently and accurately approximate the nonlinear parameter-to-solution maps of two prototypical PDEs arising in physical science and engineering applications: viscous Burgers' equation and a variable coefficient elliptic equation.

We study the Bayesian inverse problem of learning a linear operator on a Hilbert space from its noisy pointwise evaluations on random input data. Our framework assumes that this target operator is selfadjoint and diagonal in a basis shared with the Gaussian prior and noise covariance operators arising from the imposed statistical model and is able to handle target operators that are compact, bounded, or even unbounded. We establish posterior contraction rates with respect to a family of Bochner norms as the number of data tend to infinity and derive related lower bounds on the estimation error. In the large data limit, we also provide asymptotic convergence rates of suitably defined excess risk and generalization gap functionals associated with the posterior mean point estimator. In doing so, we connect the posterior consistency results to nonparametric learning theory. Furthermore, these convergence rates highlight and quantify the difficulty of learning unbounded linear operators in comparison with the learning of bounded or compact ones. Numerical experiments confirm the theory and demonstrate that similar conclusions may be expected in more general problem settings.

Despite the large number of studies of intraventricular filling dynamics for potential clinical applications, little is known as to how the diastolic vortex ring properties are altered with reduction in internal volume of the cardiac left ventricle (LV). The latter is of particular importance in LV diastolic dysfunction and in congenital diseases such as hypertrophic cardiomyopathy (HCM), where LV hypertrophy can reduce LV internal volume. We hypothesized that peak circulation and the rate of decay of circulation of the diastolic vortex would be altered with reducing end diastolic volume (EDV) due to increasing confinement. We tested this hypothesis on physical models of normal LV and HCM geometries, under identical prescribed inflow profiles and for multiple EDVs, using time-resolved particle image velocimetry measurements on a left heart simulator. Formation and pinch-off of the vortex ring were nearly unaffected with changes to geometry and EDV. Pinch-off occurred before the end of early filling (E-wave) in all test conditions. Peak circulation of the vortex core near the LV outflow tract increased with lowering EDV and was lowest for the HCM model. The rate of decay of normalized circulation in dimensionless formation time (T*) increased with decreasing EDV. When using a modified version of T* that included average LV cross-sectional area and EDV, normalized circulation of all EDVs collapsed closely in the normal LV model (10% maximum difference between EDVs). Collectively, our results show that LV shape and internal volume play a critical role in diastolic vortex ring dynamics.

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