Adaptive Causal Physics-Informed Neural Networks for Long-Horizon Dynamical System Prediction

Physics-informed neural networks (PINNs) face challenges in long-horizon temporal prediction due to the difficulty of effectively enforcing temporal causality during training. We develop a causal physics-informed framework with three complementary components. First, we introduce an inverse power-law causal weighting scheme that provides flexible control over temporal loss weighting through a single exponent. Second, we develop a learnable exponent network that dynamically adapts this exponent using training loss statistics, eliminating manual tuning. Third, we incorporate Mamba state space models as temporal encoders and develop a physics-informed fusion architecture that combines MLP and Mamba representations through causal residual training. We evaluate the proposed approaches on the Allen–Cahn equation, the chaotic Lorenz system, and two-dimensional incompressible Navier–Stokes flow. The learnable causal exponent reduces the relative $({\ell }_2)$ error by 97.6 % on Allen–Cahn, while the Mamba architecture achieves a 96.8 % reduction. For the Lorenz system, Mamba attains a minimum physics loss of $(3.30\times 10^{-7})$ at Window 4, with component-wise errors below $(1.4\times 10^{-4})$ throughout all 5 windows. On Navier–Stokes, Path 1 (LA v2) is the stronger method at equivalent compute; the physics-informed fusion PINN provides a further 6.3% error reduction over the standalone Mamba prediction at Window 3. These results demonstrate the effectiveness of adaptive causality and heterogeneous temporal representations for accurate long-horizon physics-informed learning.