用有限差分物理约束和时间步长优化离散损失以求解PDE
Optimize discrete loss with finite-difference physics constraint and time-stepping for PDE solving
AI总结：
本文提出FDTO方法，通过有限差分时间步长损失优化求解PDE，解决PINN在体符合几何和时间依赖问题上的局限，减少内存并提高精度。
摘要：
计算流体动力学（CFD）是分析流体现象和预测工程相关量的重要方法。物理规律被表述为偏微分方程（PDEs）并数值求解于计算网格上。物理指导神经网络（PINNs）作为一种流行的优化方法求解PDEs，但常常面临目标函数病态和自动微分成本高的问题。优化基于的离散化方法如ODIL通过直接优化离散变量缓解了PINN的缺点，但在体符合几何和时间依赖问题上仍存在精度和效率的限制。本文提出FDTO，一种有限差分时间步长损失优化求解器，通过离散残差定义物理损失。FDTO结合曲线坐标变换与体符合结构网格，并将长时间演化分解为序列、条件良好的子问题，与时间推进一致。该方法主要在不可压缩纳维-斯托克斯流上进行评估，包括侧壁驱动腔体基准、外部机翼气动力学（升力/阻力一致性）以及多块结构网格上的圆柱案例，具有跨块一致解。额外的验证在扩散和流体混合问题上进一步证明了通用性。与代表性的PINN求解器相比，FDTO在侧壁驱动腔体案例中减少了约82.6%的GPU内存，并在流体混合问题中实现了3-5倍更低的相对误差。这些结果表明，FDTO能够实现准确、稳定且内存高效的离散损失优化，适用于不可压缩流体解，同时适用于其他PDE模型。
英文摘要：
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.
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