Learning on the Temporal Tangent Bundle for Physics-Informed Neural Networks
This paper addresses the limitations of Physics-Informed Neural Networks for time-dependent problems by introducing a tangent bundle learning framework. Instead of directly approximating the solution, we parameterize its temporal derivative and reconstruct the state through a Volterra integral operator that enforces initial conditions exactly. This approach eliminates competing soft constraints and naturally amplifies high-frequency errors through differentiation, countering spectral bias. We prove theoretical equivalence between minimizing the differentiated residual and solving the original partial differential equation. Experiments on advection, Burgers, and Klein-Gordon equations show that the proposed method achieves 100 to 200 times lower errors than standard approaches using compact three-layer networks, with superior shock-capturing and long-time accuracy.