Continuous PDE Dynamics Forecasting with Implicit Neural Representations

TL;DR

The paper introduces DINo, a space-time continuous framework for data-driven PDE forecasting that encodes spatial observations with implicit neural representations and evolves a latent state with a learned ODE, enabling predictions at arbitrary coordinates and horizons. By separating time and space through amplitude-modulated INRs and a latent dynamics model, DINo achieves strong generalization to new grids, resolutions, and manifolds while maintaining computational efficiency. The approach outperforms state-of-the-art baselines across diverse PDEs, including wave, Navier–Stokes, and spherical shallow-water systems, especially in challenging extrapolation and irregular-grid scenarios. The work demonstrates the practicality of continuous spatiotemporal forecasting and provides a scalable path toward real-world applications such as weather forecasting and climate modeling. Future directions include scaling to larger, real-world datasets and integrating dynamics change awareness.

Abstract

Effective data-driven PDE forecasting methods often rely on fixed spatial and / or temporal discretizations. This raises limitations in real-world applications like weather prediction where flexible extrapolation at arbitrary spatiotemporal locations is required. We address this problem by introducing a new data-driven approach, DINo, that models a PDE's flow with continuous-time dynamics of spatially continuous functions. This is achieved by embedding spatial observations independently of their discretization via Implicit Neural Representations in a small latent space temporally driven by a learned ODE. This separate and flexible treatment of time and space makes DINo the first data-driven model to combine the following advantages. It extrapolates at arbitrary spatial and temporal locations; it can learn from sparse irregular grids or manifolds; at test time, it generalizes to new grids or resolutions. DINo outperforms alternative neural PDE forecasters in a variety of challenging generalization scenarios on representative PDE systems.

Figures (10)

• Figure 1: (Left) We represent time contexts. The train trajectory consists of training snapshots ([-3]$\blacksquare$), observed in a train interval $[0,T]$ denoted In-t. The line (---) in continuation is a forecasting of this trajectory beyond In-t, in $(T,T']$ denoted Out-t. The line below (---, test) is a forecasting from a new initial condition $v_0$ ([-3]$\blacksquare$) on In-t and Out-t. (Middle and right) We illustrate spatial contexts. (Middle) Dots ($\bullet$) correspond to the train observation grid ${\mathcal{X}}_{\text{tr}}$, denoted In-s. Out-s denotes the complementary domain $\Omega\setminus{\mathcal{X}}_{\text{tr}}$. (Right) New test observation grid ${\mathcal{X}}_{\text{ts}}$, used as an initial point for forecasting (left).
• Figure 2: Proposed DINo model. Inference (left): given a new initial condition observed on a grid ${\mathcal{X}}_{\text{ts}}$, $.*\vert{v_0}_{{\mathcal{X}}_{\text{ts}}}$, forecasting amounts at decoding $\alpha_t$ to $\tilde{v}_t$, by unrolling $\alpha_0$ with a time-continuous ODE dynamics model $f_\psi$. Train (right): given an observation grid ${\mathcal{X}}_{\text{tr}}$ and a space-continuous decoder $D_\phi$, $\alpha_t$ is learned by auto-decoding s.t. $D_\phi(\alpha_t)|_{{\mathcal{X}}_{\text{tr}}}=.*\vert{v_t}_{{\mathcal{X}}_{\text{tr}}}$; its evolution is then modeled with $f_\psi$.
• Figure 3: Decoding via INR Eq. (\ref{['eq:emission']}).
• Figure 4: Amplitude modulation -- Eq. (\ref{['eq:shift_modul']}). $z_t^{(l-1)}$ is input to the $l^{\text{th}}$ linear layer and combined with the spatial basis $s_{\omega^{(l)}}$ via Hadamard product.
• Figure 5: Data on manifold. DINo's Shallow-Water superresolution test prediction (top) against the reference (middle), with test MSE ($\downarrow$) (bottom).