FlowPINN: Physics-Informed Neural Flow Maps for One-Step Generation

Conditional flow matching (CFM) learns a velocity field and then generates every sample with a numerical ODE solve. We replace that pair with a single object: a neural *flow map* ${\varphi }_t(x;\theta )$ whose velocity is recovered by automatic differentiation in $t$ and which is trained by minimizing the residual of the flow ODE once, in the manner of a physics-informed neural network (PINN). Generation becomes a single network evaluation ${\varphi }_1(x_0;\theta )$, $x_0\sim p_0$. The integration work moves from sampling time, where it recurs for every sample, to training time, where it is paid once. The construction turns on one distinction that is easy to get wrong — flow matching supplies an unbiased velocity target at the network's *input*, whereas the flow ODE constrains the velocity at its *output*. Two naive substitutions of a flow map into the flow-matching loss that have exactly computable and degenerate minimizers turn to be wrong. We resolve the issue by incorporating the flow ODE residual at network output into the loss, penalized by the terminal conditional at distribution level, and hence propose a physics-informed flow matching based on PINN style solving (FlowPINN). Using benchmark examples of checkerboard and eight-Gaussian, and heavy-tailed funnel distributions, we demonstrate that one-step generation by FlowPINN reaches the accuracy (measured by energy distance) that is worth multiple steps of Euler solver or Runge-Kutta method in the classical CFM. In high-dimensional image generation problems (MNIST and CIFAR-10), FlowPINN matches a hundred-step solve to within the hardware noise band while sampling two orders of magnitude faster, and outperforms the state-of-the-art (SOTA) models including shortcut and meanflow.