Learning on the Temporal Tangent Bundle for Physics-Informed Neural Networks

TL;DR

This paper introduces a tangent bundle learning framework for physics-informed neural networks that improves accuracy and shock-capturing in time-dependent PDEs by parameterizing derivatives and reconstructing states.

Contribution

It proposes a novel tangent bundle approach that enforces initial conditions exactly and counters spectral bias, with proven theoretical equivalence to original PDE solutions.

Findings

Achieves 100 to 200 times lower errors than standard methods.

Demonstrates superior shock-capturing and long-time accuracy.

Effectively handles advection, Burgers, and Klein-Gordon equations.

Abstract

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.