Regressive-Dual PINN: Faster and More Reliable Training for Long-Horizon PDEs.

Physics-informed neural networks (PINNs) use reverse-mode automatic differentiation (AD) to solve partial differential equations (PDEs), and are expected to run fast on advanced AI hardware. In practice, their speed is far below expectations, and reliability over long time horizons is poor, often converging to trivial solutions. Prior fixes, such as finite-difference (FD) schemes and window-based training, either sacrifice accuracy or ignore the relation between successive PINN models. To address these shortcomings, we propose Regressive Dual-PINN (RD-PINN), comprising two components: Dual-FD and regressive initialization. Dual-FD computes exact first-order derivatives via dual numbers and second-order derivatives via FD, significantly reducing pure-FD's approximation error while avoiding AD's costly nested backward passes. Regressive initialization exploits the trajectory of models trained across earlier time windows, using regression to initialize a new window's parameters. Across seven PDE problems and 10 architectures, Dual-FD shows mixed speedups on one-dimensional problems. In 24 paired two-dimensional comparisons, its median speedup is 2.42× and its maximum is 9.68×; peak memory is lower in 21 of the 22 comparisons. Regressive initialization's advantage emerged on five unsteady PDE problems with extended horizons: only RD-PINN consistently sustained correct long-horizon behavior without reference data, while four published window-based and the short-horizon methods degraded as dimensionality and horizon length grew.