Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs

TL;DR

The paper introduces PI-Latent-NO, a physics-informed latent neural operator framework that learns PDE solution operators in a compact latent space using two coupled DeepONets (Latent-DeepONet and Reconstruction-DeepONet). By enforcing PDE constraints through automatic differentiation and exploiting time–space separability, the method achieves near-linear scaling and data-efficient training without requiring labeled trajectories. Across four benchmark PDEs (1D and 2D diffusion–reaction and Burgers’ equations, with variable source geometries), PI-Latent-NO matches or surpasses the accuracy of physics-informed Vanilla-DeepONet while significantly reducing training time and memory usage, and it demonstrates favorable breakeven behavior for real-time or many-query scenarios. The work highlights latent-space physics-informed learning as a promising path for scalable, real-time PDE surrogates, while outlining future improvements in latent temporal coupling, separable trunk designs, and uncertainty quantification.

Abstract

Deep operator network (DeepONet) has shown significant promise as surrogate models for systems governed by partial differential equations (PDEs), enabling accurate mappings between infinite-dimensional function spaces. However, when applied to systems with high-dimensional input-output mappings arising from large numbers of spatial and temporal collocation points, these models often require heavily overparameterized networks, leading to long training times. Latent DeepONet addresses some of these challenges by introducing a two-step approach: first learning a reduced latent space using a separate model, followed by operator learning within this latent space. While efficient, this method is inherently data-driven and lacks mechanisms for incorporating physical laws, limiting its robustness and generalizability in data-scarce settings. In this work, we propose PI-Latent-NO, a physics-informed latent neural operator framework that integrates governing physics directly into the learning process. Our architecture features two coupled DeepONets trained end-to-end: a Latent-DeepONet that learns a low-dimensional representation of the solution, and a Reconstruction-DeepONet that maps this latent representation back to the physical space. By embedding PDE constraints into the training via automatic differentiation, our method eliminates the need for labeled training data and ensures physics-consistent predictions. The proposed framework is both memory and compute-efficient, exhibiting near-constant scaling with problem size and demonstrating significant speedups over traditional physics-informed operator models. We validate our approach on a range of parametric PDEs, showcasing its accuracy, scalability, and suitability for real-time prediction in complex physical systems.

Figures (23)

• Figure 1: Schematics of (a) the baseline physics-informed vanilla neural operator (PI-Vanilla- NO) architecture and (b) our proposed physics-informed latent neural operator (PI-Latent-NO) architecture featuring coupled DeepONets for latent representation and solution reconstruction.
• Figure 2: Schematic comparing the number of trunk network evaluations between the baseline PI-Vanilla-NO and the proposed PI-Latent-NO.In a scenario requiring solution evaluation at $5$ time stamps and $10$ spatial grid points, PI-Vanilla-NO has to perform $50$ evaluations (shown in red) per input field — one for each spatiotemporal location. In contrast, PI-Latent-NO reduces the number of trunk evaluations to to just $15$ ($= 5+10$), because of its inherent separability.
• Figure 3: 1D Diffusion-reaction dynamics: Comparison between the PI-Vanilla-NO and the PI-Latent-NO (a) mean $R^2$ score of the test data, (b) mean relative $L_2$ error of test data, and (c) training per iteration. The results are based on $5$ independent runs with different seeds, varying the number of training samples $n_{\text{train}}$.
• Figure 4: 1D Diffusion-reaction dynamics: Comparison of the train and test losses with respect to runtime for models trained in a purely physics-informed manner (i.e., $n_{\text{train}} = 0$).
• Figure 5: 1D Diffusion-reaction dynamics: Comparison of all models on a representative test sample, trained in a purely physics-informed manner (i.e., $n_{\text{train}} = 0$).