Disentangled Latent Dynamics Manifold Fusion for Solving Parameterized PDEs

Abstract

Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous-time latent methods often rely on expensive test-time auto-decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics-informed framework that explicitly separates space, time, and parameters. Instead of unstable auto-decoding, DLDMF maps PDE parameters directly to a continuous latent embedding through a feed-forward network. This embedding initializes and conditions a latent state whose evolution is governed by a parameter-conditioned Neural ODE. We further introduce a dynamic manifold fusion mechanism that uses a shared decoder to combine spatial coordinates, parameter embeddings, and time-evolving latent states to reconstruct the corresponding spatiotemporal solution. By modeling prediction as latent dynamic evolution rather than static coordinate fitting, DLDMF reduces interference between parameter variation and temporal evolution while preserving a smooth and coherent solution manifold. As a result, it performs well on unseen parameter settings and in long-term temporal extrapolation. Experiments on several benchmark problems show that DLDMF consistently outperforms state-of-the-art baselines in accuracy, parameter generalization, and extrapolation robustness.

Figures (4)

• Figure 1: Physics-informed neural PDE solvers. Colorful squares denote spatial ($x$, yellow), temporal ($t$, orange), parameter ($\mu$, purple), or initial observation ($\phi$, brown) inputs. (a) PINNs process concatenated inputs directly through a decoder. (b) MAD learns a latent state $z$ for specific parameters via test-time auto-decoding. (c) PIDO relies on auto-decoding the initial observations $\phi$ to obtain a state $z_0$, then unrolls latent dynamics conditioned on $\mu$. (d) DLDMF (ours) explicitly disentangles representations by passing $x$, $t$, and $\mu$ through separate feed-forward encoders, completely avoiding iterative auto-decoding. The temporal branch unrolls the latent state $z_0$ into continuous trajectories $z_t$ via a dynamics model. A dedicated fusion block then integrates spatial ($h_x$), temporal ($z_t$), and parameter ($h_\mu$) embeddings before decoding. This design intrinsically supports long-term temporal extrapolation ("Extra horizon") and enables rapid generalization to varying PDE coefficients.
• Figure 2: Comparison of ground truth (solid black) and P$^2$INN prediction (dashed red) for $u(x,t)$ versus $x$ at $t=1.0$, $t=5.0$, and $t=10.0$.
• Figure 3: Growth of P$^2$INN's $L_2$ relative error over the time horizon $T$.
• Figure 4: P$^2$INN's error metrics versus time horizon $T$. (a) $L_2$ absolute error, (b) $L_2$ relative error, (c) maximum error grow approximately linearly beyond the training limit; (d) explained variance decreases approximately linearly.