TINNs: Time-Induced Neural Networks for Solving Time-Dependent PDEs
This addresses a bottleneck in neural network-based PDE solvers for researchers and engineers in computational physics, offering a novel method for improved accuracy and efficiency.
The paper tackled the problem of standard physics-informed neural networks (PINNs) degrading accuracy and training stability when solving time-dependent PDEs by coupling spatial features across all times, and proposed Time-Induced Neural Networks (TINNs) that parameterize network weights as a function of time, resulting in up to 4× improved accuracy and 10× faster convergence compared to PINNs.
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.