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

TL;DR

This paper introduces FDTO, a finite-difference time-stepping loss-optimization method for PDE solving that improves accuracy, stability, and memory efficiency in CFD applications by coupling coordinate transforms with structured grids.

Contribution

FDTO couples curvilinear coordinate transforms with structured grids and decomposes evolution into subproblems, advancing discrete-loss optimization for PDEs beyond PINNs.

Findings

FDTO reduces GPU memory usage by 82.6% on cavity flow.

FDTO achieves 3-5 times lower error on flow-mixing problems.

FDTO is effective for incompressible Navier-Stokes and other PDEs.

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.