Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks

In this paper, we adopt conformal prediction, a distribution-free uncertainty quantification (UQ) framework, to obtain confidence prediction intervals with coverage guarantees for Deep Operator Network (DeepONet) regression. Initially, we enhance the uncertainty quantification frameworks (B-DeepONet and Prob-DeepONet) previously proposed by the authors by using split conformal prediction. By combining conformal prediction with our Prob- and B-DeepONets, we effectively quantify uncertainty by generating rigorous confidence intervals for DeepONet prediction. Additionally, we design a novel Quantile-DeepONet that allows for a more natural use of split conformal prediction. We refer to this distribution-free effective uncertainty quantification framework as split conformal Quantile-DeepONet regression. Finally, we demonstrate the effectiveness of the proposed methods using various ordinary, partial differential equation numerical examples, and multi-fidelity learning.

B-LSTM-MIONet: Bayesian LSTM-based Neural Operators for Learning the Response of Complex Dynamical Systems to Length-Variant Multiple Input Functions

Deep Operator Network (DeepONet) is a neural network framework for learning nonlinear operators such as those from ordinary differential equations (ODEs) describing complex systems. Multiple-input deep neural operators (MIONet) extended DeepONet to allow multiple input functions in different Banach spaces. MIONet offers flexibility in training dataset grid spacing, without constraints on output location. However, it requires offline inputs and cannot handle varying sequence lengths in testing datasets, limiting its real-time application in dynamic complex systems. This work redesigns MIONet, integrating Long Short Term Memory (LSTM) to learn neural operators from time-dependent data. This approach overcomes data discretization constraints and harnesses LSTM's capability with variable-length, real-time data. Factors affecting learning performance, like algorithm extrapolation ability are presented. The framework is enhanced with uncertainty quantification through a novel Bayesian method, sampling from MIONet parameter distributions. Consequently, we develop the B-LSTM-MIONet, incorporating LSTM's temporal strengths with Bayesian robustness, resulting in a more precise and reliable model for noisy datasets.

A Physics-Guided Bi-Fidelity Fourier-Featured Operator Learning Framework for Predicting Time Evolution of Drag and Lift Coefficients

In the pursuit of accurate experimental and computational data while minimizing effort, there is a constant need for high-fidelity results. However, achieving such results often requires significant computational resources. To address this challenge, this paper proposes a deep operator learning-based framework that requires a limited high-fidelity dataset for training. We introduce a novel physics-guided, bi-fidelity, Fourier-featured Deep Operator Network (DeepONet) framework that effectively combines low and high-fidelity datasets, leveraging the strengths of each. In our methodology, we began by designing a physics-guided Fourier-featured DeepONet, drawing inspiration from the intrinsic physical behavior of the target solution. Subsequently, we train this network to primarily learn the low-fidelity solution, utilizing an extensive dataset. This process ensures a comprehensive grasp of the foundational solution patterns. Following this foundational learning, the low-fidelity deep operator network's output is enhanced using a physics-guided Fourier-featured residual deep operator network. This network refines the initial low-fidelity output, achieving the high-fidelity solution by employing a small high-fidelity dataset for training. Notably, in our framework, we employ the Fourier feature network as the Trunk network for the DeepONets, given its proficiency in capturing and learning the oscillatory nature of the target solution with high precision. We validate our approach using a well-known 2D benchmark cylinder problem, which aims to predict the time trajectories of lift and drag coefficients. The results highlight that the physics-guided Fourier-featured deep operator network, serving as a foundational building block of our framework, possesses superior predictive capability for the lift and drag coefficients compared to its data-driven counterparts.

Deep Operator Learning-based Surrogate Models with Uncertainty Quantification for Optimizing Internal Cooling Channel Rib Profiles

This paper designs surrogate models with uncertainty quantification capabilities to improve the thermal performance of rib-turbulated internal cooling channels effectively. To construct the surrogate, we use the deep operator network (DeepONet) framework, a novel class of neural networks designed to approximate mappings between infinite-dimensional spaces using relatively small datasets. The proposed DeepONet takes an arbitrary continuous rib geometry with control points as input and outputs continuous detailed information about the distribution of pressure and heat transfer around the profiled ribs. The datasets needed to train and test the proposed DeepONet framework were obtained by simulating a 2D rib-roughened internal cooling channel. To accomplish this, we continuously modified the input rib geometry by adjusting the control points according to a simple random distribution with constraints, rather than following a predefined path or sampling method. The studied channel has a hydraulic diameter, Dh, of 66.7 mm, and a length-to-hydraulic diameter ratio, L/Dh, of 10. The ratio of rib center height to hydraulic diameter (e/Dh), which was not changed during the rib profile update, was maintained at a constant value of 0.048. The ribs were placed in the channel with a pitch-to-height ratio (P/e) of 10. In addition, we provide the proposed surrogates with effective uncertainty quantification capabilities. This is achieved by converting the DeepONet framework into a Bayesian DeepONet (B-DeepONet). B-DeepONet samples from the posterior distribution of DeepONet parameters using the novel framework of stochastic gradient replica-exchange MCMC.