Optimize discrete loss with finite-difference physics constraint and time-stepping for solving incompressible flow

TL;DR

FDTO is proposed, a finite-difference time-stepping loss-optimization solver that defines physics losses from discrete residuals and enables accurate, stable, and memory-efficient discrete-loss optimization for incompressible-flow solutions, while remaining applicable to other PDE models.

Abstract

Computational Fluid Dynamics (CFD) is an important approach for analyzing flow phenomena and predicting engineering-relevant quantities. The governing physics is formulated as partial differential equations(PDEs) and solved numerically on computational grids. Physics-informed neural networks(PINNs) have emerged as a popular optimization-based approach for solving PDEs, but they often suffer from ill-conditioned objectives and the high cost of automatic differentiation. Optimization-based discretizations such as ODIL mitigate several PINN drawbacks by optimizing discrete variables directly, yet accuracy and efficiency remain limited on body-fitted geometries and for time-dependent problems. This paper proposes FDTO, a finite-difference time-stepping loss-optimization solver that defines physics losses from discrete residuals. FDTO couples curvilinear coordinate transforms with body-fitted structured grids and decomposes long-horizon evolution into sequential, well-conditioned subproblems consistent with time marching. The method is primarily evaluated on incompressible Navier-Stokes flows, including lid-driven cavity benchmarks, external airfoil aerodynamics (lift/drag consistency), and a cylinder case on a multi-block structured mesh with cross-block coherent solutions. Additional validations on diffusion and flow-mixing problems further demonstrate generality. Compared with representative PINN-based solvers, FDTO reduces GPU memory by about 82.6% on the lid-driven cavity case and achieves 3-5 times lower relative error on the flow-mixing problem. These results indicate that FDTO enables accurate, stable, and memory-efficient discrete-loss optimization for incompressible-flow solutions, while remaining applicable to other PDE models.

Figures (27)

• Figure 1: Paradigms of PDE solvers. top: CFD; middle: FDTO; bottom: PINN.
• Figure 2: An example of coordinate transformation from physical space (left) to computational space (right).
• Figure 3: Diagram of FDTO spatial discretization procedure. (left) Node-stored variables (and precomputed grid metrics) are linearly interpolated to edge midpoints to obtain face states. (middle) Conservative numerical fluxes are evaluated on these faces using the interpolated states. (right) The nodal residual is assembled by a finite-difference flux divergence, i.e., differences of neighboring face fluxes in the computational $\xi$/$\eta$ directions.
• Figure 4: Schematic of N-C-N averaging operator on body-fitted meshes.
• Figure 5: Boundary Conditions for the lid-driven cavity (Uniform).