Parameterized Physics-informed Neural Networks for Parameterized PDEs

TL;DR

This work addresses the challenge of efficiently solving parameterized PDEs by introducing P$^2$INNs, which explicitly encode PDE parameters into a latent representation and combine them with spatiotemporal coordinates through a three-part architecture (two encoders and a manifold network). The model is trained across multiple parameter configurations in a single run, achieving higher accuracy and parameter efficiency than traditional PINNs, and mitigating common failure modes on 1D Convection-Diffusion-Reaction and 2D Helmholtz benchmarks. A fast fine-tuning strategy based on SVD modulation enables rapid adaptation to new PDE parameters with limited fine-tuning. Overall, P$^2$INNs offer robust, scalable surrogate modeling for parameterized PDEs with strong potential for real-time design and uncertainty analysis.

Abstract

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Reynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P$^2$INNs). P$^2$INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P$^2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known "failure modes".

Figures (14)

• Figure 1: P$^2$INNs outperform the baselines. P$^2$INNs reduce the average $L_2$ absolute (Abs.) and relative (Rel.) errors by 100$\times$ compared to the baselines. The results are for reaction equations -- the most challenging problems for PINNs.
• Figure 2: The ground-truth solutions of various convection equations with an initial condition of $1 + \sin(x)$ (Figure \ref{['fig:figure1']}. (a)-(c)) and reaction equations with an initial condition of a Gaussian distribution $N(\pi, ({\pi}/2)^2)$ (Figure \ref{['fig:figure1']}. (d)-(f)). We note that varied solutions are made (with similar architectures) depending on changes in coefficient.
• Figure 3: The ground-truth solutions of various CDR equations with an initial condition of $1 + \sin(x)$ (Figure \ref{['fig:figure2']}. (a)-(c)) or a Gaussian distribution $N(\pi, ({\pi}/2)^2)$ (Figure \ref{['fig:figure2']}. (d)-(f)). We note that the solution in the last column reflects the first two columns' solutions. Therefore, there also exist similarities across different equation types.
• Figure 4: P$^2$INNs architecture. The two encoders $g_{\theta_p}$ and $g_{\theta_c}$ are added to generate better representations for the PDE parameter and the spatial/temporal coordinate. We also customize the manifold network $g_{\theta_g}$. In this figure, we provide the CDR equation as an example.
• Figure 5: P$^2$INNs with SVD modulation. From the pre-trained decoder layer of P$^2$INN, we obtain the bases $\Phi_l, \Psi_l$ for parameterized PDEs through SVD (cf. Eq. \ref{['eq:eq_svd_mod']}). Note that only the diagonal matrices $\alpha_l$ are used for fine-tuning. (The dotted lines represent learnable parameters.)