TINNs: Time-Induced Neural Networks for Solving Time-Dependent PDEs

TL;DR

TINNs introduce a novel neural network architecture that models the evolution of spatial features over time for solving PDEs, leading to significant improvements in accuracy and convergence speed over traditional PINNs.

Contribution

The paper proposes Time-Induced Neural Networks (TINNs), which parameterize network weights as functions of time, enabling better modeling of dynamic PDE solutions.

Findings

Up to 4x improved accuracy over PINNs.

Up to 10x faster convergence.

Effective handling of time-dependent PDEs.

Abstract

Physics-informed neural networks (PINNs) solve time-dependent partial differential equations (PDEs) by learning a mesh-free, differentiable solution that can be evaluated anywhere in space and time. However, standard space--time PINNs take time as an input but reuse a single network with shared weights across all times, forcing the same features to represent markedly different dynamics. This coupling degrades accuracy and can destabilize training when enforcing PDE, boundary, and initial constraints jointly. We propose Time-Induced Neural Networks (TINNs), a novel architecture that parameterizes the network weights as a learned function of time, allowing the effective spatial representation to evolve over time while maintaining shared structure. The resulting formulation naturally yields a nonlinear least-squares problem, which we optimize efficiently using a Levenberg--Marquardt method. Experiments on various time-dependent PDEs show up to $4\times$ improved accuracy and $10\times$ faster convergence compared to PINNs and strong baselines.