arXiv:2511.18820physics.flu-dyncs.LG2025-11

无需标注数据,用神经网络精准模拟高雷诺数流体,自动生成涡旋。

Unsupervised simulation of incompressible flows with physics- and equality- constrained artificial neural networks

  • 基于压力-泊松方程构建目标函数,严格满足无散度与边界条件。
  • 在 $Re=7,500$ 的腔道流和圆柱绕流中实现稳定求解,捕捉自发涡激振荡。
  • 适合追求物理精确性、不依赖标注数据的流体仿真研究者。

物理信息神经网络(PINNs)在求解偏微分方程方面展现出潜力,但在高雷诺数不可压缩流动模拟中的表现仍受限。现有方法依赖辅助标签数据、监督预训练或参考解,尚未出现可媲美传统有限差分/有限体积法的纯无监督方法。本文认为该差距源于缺乏对无散度约束与边界条件的严格强制机制。为此,采用物理与等式约束人工神经网络(PECANN)框架,结合条件自适应增广拉格朗日法(CA-ALM),并引入基于压力-泊松方程的目标函数:最小化压力泊松方程残差,同时将动量方程、连续性方程及原始变量边界条件作为等式约束,由CA-ALM严格施加。针对对流主导的高雷诺数流动,进一步提出自适应消失熵黏性,以稳定早期训练而不影响收敛解。对比实验表明,仅以动量残差为目标的基线方法在相同框架下无效,凸显压力-泊松目标的关键作用。方法在 $Re=7{,}500$ 的拖曳腔流、三维非定常贝尔特拉米流以及具有通用进/出口边界条件的圆柱定常与非定常绕流上进行了评估,并完成消融研究,识别出可接受的出口条件——全程无需标注数据或监督预训练。尤为关键的是,从随机初始化网络出发,成功捕捉了非定常圆柱绕流中自发产生的周期性涡脱落现象。

原文摘要 · Abstract (English)

Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations, yet their success in simulating incompressible flows at high Reynolds numbers remains limited. Existing approaches rely on auxiliary labeled data, supervised pretraining, or reference solutions, and no purely unsupervised method comparable to conventional finite-difference or finite-volume solvers has been demonstrated. We attribute this gap to the absence of a mechanism for enforcing the divergence-free constraint and boundary conditions to strict tolerances. To address this, we adopt the physics- and equality-constrained artificial neural network (PECANN) framework with a conditionally adaptive augmented Lagrangian method (CA-ALM), and introduce a pressure-Poisson-based objective. The residual of the pressure Poisson equation is minimized subject to the momentum and continuity equations and boundary conditions on the primitive variables as equality constraints, with CA-ALM enforcing all constraints tightly. For advection-dominated, high-Reynolds-number flows, we further propose an adaptive vanishing entropy viscosity that stabilizes early training without influencing the converged solution. A baseline that instead uses the momentum residual as the objective proves ineffective under the same machinery, underscoring the critical role of the pressure-Poisson objective. The method is assessed on lid-driven cavity flow up to $Re=7{,}500$, three-dimensional unsteady Beltrami flow, and steady and unsteady flow past a circular cylinder with general inflow-outflow boundary conditions, including an ablation study identifying admissible outlet conditions -- all without labeled data or supervised pretraining. Notably, it captures the spontaneous onset of periodic vortex shedding in unsteady cylinder flow without external perturbations, starting from a randomly initialized network.

流体模拟神经网络物理信息无监督

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