David C. Del Rey Fernández

Zelalem Arega Worku, David C. Del Rey Fernández, David W. Zingg

The Lax equivalence theorem guarantees convergence of stable and consistent discretizations for linear hyperbolic partial differential equations (PDEs). For nonlinear problems, however, stability and consistency alone do not generally guarantee convergence, even for smooth solutions, and existing convergence results typically rely either on projection-based error decompositions or on linearization arguments that do not directly extend to entropy-stable split-form discretizations. In particular, general convergence results for entropy-stable discretizations of hyperbolic PDEs are currently lacking, despite their widespread use. In this work, we prove convergence under smoothness assumptions on the exact solution and fluxes for entropy-stable split-form discretizations of scalar and symmetric hyperbolic systems with homogeneous flux functions within the continuous summation-by-parts (C-SBP) framework. The scalar inviscid Burgers equation is presented as a canonical example. The analysis is based on a stability-consistency argument that yields a nonlinear error evolution inequality whose solution provides an explicit upper bound on the numerical error. We show that, for sufficiently small mesh spacing, and for degree-$p$ C-SBP discretizations in $d$ spatial dimensions with $p>1+d/2$, this bound remains finite on any finite time interval and tends to zero as the mesh is refined, implying convergence despite the presence of local linear instabilities. The results help clarify the relationship between consistency, entropy stability, nonlinear error growth, and convergence for discretizations of nonlinear hyperbolic problems.

Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández et al.

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error. We prove that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.