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

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 relative $L^2$ error and $10\times$ faster convergence compared to PINNs and strong baselines. Code is available at https://github.com/CYDai-ml/TINN.

Predicting shifting real-world phenomena, like worsening storms or spreading heat, requires solving complex mathematical equations that change over time. Traditionally, since AI models share the same parameters across all timestamps, they struggle to capture erratic, fast-moving changes, often leading to unstable or inaccurate results. To solve this, we introduce Time-Induced Neural Networks (TINNs). Instead of keeping parameters static, TINN allows its core parameters to fluidly adapt and evolve as time progresses, acting like a brain that naturally reshapes itself to match the changing environment. This break from rigid structures improves both the accuracy and stability of tracking complex systems. Ultimately, this approach makes physical simulations more reliable, paving the way for better weather forecasting, safer aerospace engineering, and advanced climate modeling.