Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems

TL;DR

This work tackles operator learning for complex, multi-physics systems with data from multiple fidelities. It introduces a composite DeepONet architecture comprising low-fidelity, nonlinear, and linear subnetworks to capture both linear and nonlinear correlations between fidelities, with data-driven and physics-informed variants. Through one- and two-dimensional synthetic tests and ice-sheet modeling, the authors demonstrate improved accuracy and substantial cost savings when leveraging abundant low-fidelity data alongside sparse high-fidelity data or physics constraints, including multi-resolution and multi-model scenarios. The framework is versatile, robust to noise, and extensible to additional fidelities and partial physics knowledge, offering a practical path toward efficient uncertainty quantification and rapid surrogates for complex systems.

Abstract

Operator learning for complex nonlinear systems is increasingly common in modeling multi-physics and multi-scale systems. However, training such high-dimensional operators requires a large amount of expensive, high-fidelity data, either from experiments or simulations. In this work, we present a composite Deep Operator Network (DeepONet) for learning using two datasets with different levels of fidelity to accurately learn complex operators when sufficient high-fidelity data is not available. Additionally, we demonstrate that the presence of low-fidelity data can improve the predictions of physics-informed learning with DeepONets. We demonstrate the new multi-fidelity training in diverse examples, including modeling of the ice-sheet dynamics of the Humboldt glacier, Greenland, using two different fidelity models and also using the same physical model at two different resolutions.

Figures (27)

• Figure 1: Schematic of the composite physics-informed multifidelity DeepONet setup. $\mathcal{F}_{LF}({u})(x),$$\mathcal{F}_{nl}({u})(x)$, and $\mathcal{F}_{l}({u})(x)$ are the outputs of the low-fidelity, nonlinear, and linear DeepONet subnets.
• Figure 2: Data-driven multifidelity: one-dimensional, jump function. (a-b) Results of the single fidelity and multifidelity predictions of the high- and low-fidelity data. (c) Single-fidelity error as a function of $a$ and $x$, (d) multifidelity high-fidelity prediction error as a function of $a$ and $x$, and (e) multifidelity low-fidelity prediction error as a function of $a$ and $x$.
• Figure 3: Data-driven multifidelity: two-dimensional, nonlinear correlation. (a) Absolute error of the high-fidelity prediction, multifidelity prediction of the high-fidelity data, and multifidelity prediction of the low-fidelity data for $a = 8.5211$. (b) Absolute error of the high-fidelity prediction, multifidelity prediction of the high-fidelity data, and multifidelity prediction of the low-fidelity data for $a = 9.5737$. The high-fidelity data points are shown in white for clarity.
• Figure 4: Data-driven multifidelity: two-dimensional nonlinear correlation. From left to right: exact high-fidelity solution, multifidelity DeepONet prediction, multifidelity DeepONet nonlinear correlation, and multifidelity DeepONet linear correlation. The high-fidelity data points are shown in white for clarity.
• Figure 5: Data-driven multifidelity: multiresolution ice-sheet dynamics. Example of the ice-sheet dynamics from an MOLHO simulation with resolution 41x41 at times $t = 1.0$ yr and $t = 50.0$ yr. The basal friction is denoted by $\beta$ (units: [Pa yr / m]), the ice thickness by $H$ (units: [m]), and the $x-$ and $y-$ depth-averaged velocities by $u$ and $v$ (units: [m / yr]). The basal friction $\beta$ is constant in time.