Enhancing physical consistency in stochastic optimization for adjoint-based inverse problems: Application to compressible RANS simulations in the discontinuous Galerkin framework

• Deterministic and stochastic methods are studied for adjoint-based inverse modeling. • Novel stochastic update rule preserves boundary and regularity via adjoint-state. • When coupled with high-order schemes, it enhances smoothness of tuning parameter. • New method outperforms L-BFGS and N-ADAM in multi-objective inverse problems. This work explores the use of quasi-Newton and stochastic optimization methods for gradient-based inverse problems in physical modeling, focusing on their application within high-order numerical frameworks. Such problems are often characterized by the following challenges: (i) physical constraints often yield a highly non-convex design space with multiple local optima; (ii) solutions must satisfy intrinsic properties, such as boundary and regularity conditions, which are not easily enforced as explicit constraints. Conventional stochastic optimizers, such as N-ADAM, exhibit significant information loss in their update rules, leading to non-physical solutions. To overcome these limitations, we introduce a novel stochastic optimizer, V-N-ADAM-DG, which incorporates adjoint-state information into the update rule to maintain physically meaningful corrections in terms of regularity and boundary conditions. We validate our approach in the context of mean-flow reconstruction for Reynolds-averaged Navier-Stokes (RANS) simulations using a high-order discontinuous Galerkin (DG) discretization method, as proposed by Fanizza et al. (2025). The optimization framework considers both vectorial corrective terms, inferred in the momentum and energy equations, and scalar corrective terms in the Spalart-Allmaras (SA) transport equation. The V-N-ADAM-DG optimizer effectively reconstructs mean flow quantities while ensuring smooth transitions of corrective parameters at boundaries, an improvement over standard stochastic optimizers. Additionally, it facilitates a rapid decay of the optimal degrees of freedom (DOFs), leading to smoother corrections in high-order reconstructions-achieving a balance between the robustness of quasi-Newton methods (such as L-BFGS) and the flexibility of stochastic approaches. Numerical experiments across various flow configurations demonstrate that V-N-ADAM-DG consistently outperforms both L-BFGS and N-ADAM, particularly in complex inverse problems that employ multiple combined cost functions to reconstruct different physical quantities.