We present a method for estimating solutions to partial differential equations with uncertain parameters using a modification of the Bayesian Multivariate Adaptive Regression Splines (BMARS) emulator. The BMARS algorithm uses Markov chain Monte Carlo (MCMC) to construct a basis function composed of polynomial spline functions, for which derivatives and integrals are straightforward to compute. We use these calculations and a modification of the curve-fitting BMARS algorithm to search for a basis function (response surface) which, in combination with its derivatives/integrals, satisfies a governing differential equation and specified boundary condition. We further show that this fit can be improved by enforcing a conservation or other physics-based constraint.
Our results indicate that estimates to solutions of simple first-order partial differential equations (without uncertainty) can be efficiently computed with very little regression error. We then extend the method to estimate uncertainties in the solution to a pure absorber transport problem in a medium with uncertain cross-section. We describe and compare two strategies for propagating the uncertain cross section through the BMARS algorithm; the results from each method are in close comparison with analytic results. We discuss the scalability of the algorithm to parallel architectures and the applicability of the two strategies to larger problems with more degrees of uncertainty.