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

TL;DR

The paper identifies a time-entanglement problem in standard space–time PINNs, where a single parameter vector must capture dynamics with evolving spatial complexity. It proposes Time-Induced Neural Networks (TINNs), which encode temporal evolution as a trajectory in parameter space by learning a compact time code and a layer-wise affine lift to the full parameter set, thereby decoupling temporal dynamics from the spatial representation. Training leverages a nonlinear least-squares formulation using Levenberg–Marquardt optimization, yielding faster convergence and improved robustness. Across five time-dependent PDE benchmarks, TINNs achieve higher accuracy with far fewer parameters than strong baselines, demonstrating practical gains in both performance and training stability for physics-informed learning of dynamic systems.

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.

Figures (6)

• Figure 1: Space--time PINN (time as input) vs. TINN (time in parameter space). The left shows how a vanilla neural network structure deals with time-dependent PDEs. The time is incorporated into the input dimension. On the right-hand side, the time information in TINN is integrated into the parameter space, making it easier to capture the complex dynamics of the system.
• Figure 2: Motivation for TINNs on viscous Burgers.Left: space--time contour of the exact solution on $x\in[-1,1]$, $t\in[0,1]$. Right: solution profiles at $t\in\{0,0.5,1.0\}$. The solution evolves from a smooth profile to a thin transition layer near $x=0$, producing much steeper spatial gradients at later times. Although the profiles at $t=0.5$ and $t=1.0$ are similar in shape, their sharpness differs substantially, which is challenging for a single space--time network with shared parameters and motivates TINN's time-modulated parameterizations.
• Figure 3: Absolute error of the spatial derivative at $x=0$ over $t\in[0, 1]$: vanilla MLP vs. TINN. With comparable parameter counts, TINN yields smaller and more stable errors. See details in \ref{['app: MLP vs TINN']}.
• Figure 4: Architecture for time-dependent parameters $\bm{\theta}(t)$ in TINN. A small dense network $\bm{\mathcal{N}}(t)$ outputs a $2L$-dimensional code $\bm{\Phi}(t)$, which is lifted to the full parameter vector $\bm{\theta}(t)\in\mathbb{R}^{N_D}$ via an entrywise affine map, avoiding direct prediction of an $N_D$-dimensional output.
• Figure 5: Training stability on viscous Burgers: vanilla PINN vs. TINN.Left: training loss. Right:$\ell_2$-norm of parameter updates. Despite resampling spikes, vanilla PINN shows high-variance loss and irregular parameter updates, whereas TINN converges more smoothly with more regular update magnitudes.

Theorems & Definitions (1)

• Proposition 5.1