[Paper Review] Learning Incompressible Fluid Dynamics from Scratch -- Towards Fast, Differentiable Fluid Models that Generalize

Nils Wandel, Michael Weinmann+1 more|arXiv (Cornell University)|Jun 15, 2020

Model Reduction and Neural Networks29 references33 citations

TL;DR

Presents an unsupervised, physics-constrained CNN framework that learns incompressible fluid dynamics from scratch, yielding fast, differentiable fluid solvers that generalize to new domains without training data.

ABSTRACT

Fast and stable fluid simulations are an essential prerequisite for applications ranging from computer-generated imagery to computer-aided design in research and development. However, solving the partial differential equations of incompressible fluids is a challenging task and traditional numerical approximation schemes come at high computational costs. Recent deep learning based approaches promise vast speed-ups but do not generalize to new fluid domains, require fluid simulation data for training, or rely on complex pipelines that outsource major parts of the fluid simulation to traditional methods. In this work, we propose a novel physics-constrained training approach that generalizes to new fluid domains, requires no fluid simulation data, and allows convolutional neural networks to map a fluid state from time-point t to a subsequent state at time t + dt in a single forward pass. This simplifies the pipeline to train and evaluate neural fluid models. After training, the framework yields models that are capable of fast fluid simulations and can handle various fluid phenomena including the Magnus effect and Karman vortex streets. We present an interactive real-time demo to show the speed and generalization capabilities of our trained models. Moreover, the trained neural networks are efficient differentiable fluid solvers as they offer a differentiable update step to advance the fluid simulation in time. We exploit this fact in a proof-of-concept optimal control experiment. Our models significantly outperform a recent differentiable fluid solver in terms of computational speed and accuracy.

Motivation & Objective

• Develop a physics-constrained, unsupervised training approach for incompressible fluid dynamics that does not require simulated data.
• Generalize fluid models to unseen domain geometries while maintaining physical fidelity.
• Produce fast, differentiable neural solvers capable of real-time fluid simulation.
• Demonstrate the model’s ability to capture viscous effects and phenomena like Magnus effect and Kármán vortex streets.

Proposed method

• Use Helmholtz decomposition to enforce incompressibility by predicting a vector potential a such that v = curl(a).
• Represent the 2D flow on a staggered MAC grid and discretize in time with explicit/implicit/IMEX schemes.
• Feed a feature-rich representation into a U-Net to update p and a, then update the fluid state by adding the network output to the current state.
• Train with a physics-constrained loss using Navier–Stokes residuals and a boundary loss, optionally including a pressure regularization term at high Reynolds numbers.
• Employ a recycling training strategy that updates and reuses predicted fields to generate progressively realistic fluid states without ground-truth data.
• Provide an ablation model (v-Net) that directly predicts velocity without vector potential to compare performance.

Experimental results

Research questions

• RQ1 Can a CNN learn the complete incompressible Navier–Stokes update step from scratch without training data?
• RQ2 Does enforcing incompressibility via a vector potential enable generalization to unseen domain geometries?
• RQ3 How does a physics-constrained, unsupervised training regime compare to data-driven or solver-based baselines in speed and accuracy?
• RQ4 Can the model capture viscous effects and complex phenomena such as Magnus effect and Kármán vortex streets?
• RQ5 Is the trained model differentiable enough to enable gradient-based control of fluid flows?

Key findings

• The a-Net (vector-potential-based) model achieves higher accuracy than the velocity-only ablation and outperforms PhiFlow in both divergence and momentum losses on a 100×100 test setup.
• The a-Net runs significantly faster than PhiFlow, delivering higher timestep-per-second rates on CPU (11x) and GPU (40x).
• For dt = 4, a-Net achieves L_d = 5.44e-7 and L_p = 1.56e-5, outperforming the v-Net in both metrics.
• Qualitative results show correct wake dynamics across Reynolds numbers and replication of Magnus effect and lift-generating pressure fields around airfoils not seen during training.
• The model demonstrates real-time capable simulations (~250 timesteps/s on 100×300 grid) and differentiable behavior enabling gradient-based optimal control of vortex shedding frequencies.