Physics-informed neural networks (PINNs) have gained wide attention as neural network-based solvers for partial differential equations (PDEs). They offer a flexible way to combine data with governing laws. However, in practice, these methods often face challenges in generalizing beyond the training domain, especially for time-dependent and nonlinear PDEs. This limits their reliability in real scientific and engineering applications. In this work, we present the framework to improve the generalization of physics-informed PDE solvers by explicitly exploiting the causal and sequential structure of PDE solutions. Our approach couples a physics-informed neural network, which learns the solution within the training domain, with a recurrent architecture based on systems of ordinary differential equations, referred to as neural oscillators. These models evolve their hidden states through differential equations, allowing them to capture long-time dynamics and avoid common optimization issues such as exploding and vanishing gradients. The trained neural oscillator learns the temporal evolution of the PDE solution and is then used to extrapolate beyond the training domain without access to new data. We demonstrate the method on several benchmark problems, including the viscous Burgers, Allen--Cahn, and nonlinear Schrödinger equations, as well as a fourth-order Euler--Bernoulli beam equation. Across all cases, the proposed approach consistently outperforms standard recurrent models and state-of-the-art PINN generalization techniques, achieving higher accuracy in unseen regions. These results show that embedding continuous-time dynamical structure into neural PDE solvers improves their robustness and predictive power. The work highlights how blending physics-informed learning with neural differential equation models can help bridge the gap between theory and practical deployment of neural network solvers for PDEs.