Abstract Operator networks are neural networks designed to learn operators with special emphasis on solution operators for parameterized families of partial differential equations (PDEs). Once trained, operator networks can provide a solution to a PDE more quickly than current numerical PDE solvers by several orders of magnitude. Fourier neural operators (FNOs) and deep operator networks (DeepONets) are the two primary operator networks in existence for learning the solution operator to PDEs and have mostly only been applied to two-dimensional or three-dimensional problems, due to the computational expense of training networks in higher dimensional settings. The sole exception is a model-parallel FNO, which decomposes the function input domain space. We demonstrate a neural operator network with a physics-informed integral kernel that, once trained, is able to predict skin and ocular media’s time-dependent thermal response to incident laser radiation much more rapidly than existing numerical algorithms.

Diffuse Optical Tomography (DOT) is a well-known imaging technique for detecting the optical properties of an object in order to detect anomalies, such as diffusive or absorptive targets. More specifically, DOT has many applications in medical imaging including breast cancer screening. It is an affordable and noninvasive method to recover the optical properties of a body using photon density measurements from applying laser sources placed at its surface. Mathematically, the reconstruction of the internal absorption or scattering is a severely ill-posed inverse problem and yields a poor quality image reconstruction. Studying coefficient inverse problems in a stochastic setting has increasingly gained in prominence in the past couple of decades. In this work, we will show convergence and optimality for a Bayesian estimator for the absorption coefficient built from the noisy data obtained in a simplified DOT Model. We establish the rate of convergence of such an estimator in the supremum norm loss and show that it is optimal. We also present numerical experiments in support of our theoretical findings.

Microscopic roadway traffic simulators, which attempt to mimic real-world driver behaviors, are based on car-following models and have been widely used as a cost-effective tool for intelligent transportation system (ITS) evaluation. In addition to evaluation, ITSs can benefit from accurate car-following models that can provide current estimations and future predictions of various traffic situations to support real-time traffic management. The accuracy and reliability of these applications are greatly dependent on the appropriate calibration of car-following models. In this paper, the authors developed a process to apply a stochastic calibration method with appropriate regularization to estimate the distribution of parameters for car-following models. The calibration method is based on the Markov chain Monte Carlo simulation that uses the Bayesian estimation theory. An intelligent driver model and Helly's car-following model were utilized to evaluate this method. The Bayesian approach provided better results in terms of the cost function than the deterministic optimization algorithm. With the Bayesian approach, the mean square error per vehicle is decreased with the increased number of vehicles. Analysis also revealed that the Bayesian approach predicted drivers' speed and acceleration/deceleration profiles more closely to the real-world data compared with the deterministic approach evaluated in this paper. Positive validation outcomes suggest potential efficacy of the calibration approach presented in this paper for future applications.