Abstract. Gaussian processes (GPs) are broadly used as surrogate models for expensive computer simulators of complex phenomena. However, a key bottleneck is that its training data are generated from this expensive simulator and thus can be highly limited. A promising solution is to supplement the surrogate with boundary information from scientific knowledge. However, despite recent work, existing boundary-integrated GPs face key limitations for incorporating broad boundary information on irregular (i.e., nonhypercube) domains: they do not provide approximation error analysis nor a means for enforcing sample path smoothness, both of which are important for reliable surrogates. We thus propose a novel BdryMatérn GP framework, which can reliably integrate Dirichlet, Neumann, and Robin boundary information on an irregular domain with mild smoothness conditions on its boundaries. Our model leverages a new BdryMatérn covariance kernel derived in path integral form via a stochastic partial differential equation formulation. Similar to the GP with a Matérn kernel, we prove that sample paths from the BdryMatérn GP satisfy the desired boundaries with smoothness control on its derivatives. We further present an efficient approximation procedure for the BdryMatérn kernel using finite element modeling with rigorous error analysis. Finally, we demonstrate the effectiveness of the BdryMatérn GP in a suite of simulation experiments and PDE applications that provide boundary information on irregular domains.
Abstract. We introduce a new global sensitivity measure, the global activity scores. The measure is based on finite differences of the underlying function, in contrast to several sensitivity measures in the literature that are based on derivatives of the function. We establish its theoretical connection with Sobol’ sensitivity indices and demonstrate its performance through numerical examples. In these examples, we compare global activity scores with Sobol’ sensitivity indices, derivative-based sensitivity measures, and activity scores. The results show that in the presence of additive noise or high variability, global activity scores provide more stable and reliable identification of influential variables than derivative-based measures and activity scores, which are more sensitive to noise. In noiseless settings, however, all three approaches yield comparable results.
Markov chain Monte Carlo (MCMC) requires only the ability to evaluate the likelihood, making it a common technique for inference in complex models. However, it can have a slow mixing rate, requiring the generation of many samples to obtain good estimates and an overall high computational cost. FLARE MCMC is a multifidelity layered MCMC method that exploits lower-fidelity approximations of the true likelihood calculation to improve mixing and leads to overall faster performance. Such lower-fidelity likelihoods are commonly available in scientific and engineering applications where the model involves a simulation whose resolution or accuracy can be tuned. Our technique uses recursive, layered chains with simple layer tuning; it does not require the likelihood to take any specific form or have any particular internal mathematical structure. We demonstrate experimentally that FLARE MCMC achieves larger effective sample sizes for the same computational time across different scientific domains, including hydrology and cosmology.
Motivated by challenges arising in molecular simulation, we study reactive trajectories of the over-damped Langevin dynamics, i.e., trajectories observed as they pass from a set A corresponding to the reagents of a chemical reaction to a set B corresponding to the products. Reactive trajectories are known to have the same distribution as trajectories of the overdamped Langevin dynamics biased by a singular drift related to the committor function. In this work, we assess the effect of replacing the exact singular drift with an approximation based on an approximate committor function. We derive a convenient formula for the relative entropy between the distributions of exact and approximate reactive trajectories, and we propose a stochastic gradient descent method for minimizing the entropy to train an approximate committor function on the fly while computing reactive trajectories. We also devise a model assessment procedure for comparing the qualities of different approximations to the committor function based on the relative entropy.
The Kalman(-Bucy) filter is the natural choice for the state reconstruction of disturbed, linear dynamical systems based on flawed and incomplete measurements. Taking a deterministic viewpoint, this work investigates possible extensions of the concept to systems with uncertain dynamics and noise covariances. In a theoretical analysis, error bounds in terms of the variance of the uncertainties are derived. The article concludes with a numerical implementation of two example systems, allowing for a comparison of the estimators.
