A Trainable-by-Parts Operator Learning Framework: Bridging DeepONet and Karhunen-Loeve Expansions for Large-Scale Applications
Authors: Christian Munoz, Alexandre Tartakovsky
Organizations: Civil Engineering Department University of Illinois Urbana-Champaign Urbana, IL 61801 · Pacific Northwest National Laboratory,Jun Richland, WA 99352
Training operator-learning models for large-scale problems governed by partial differential equations (PDEs) is challenging due to the curse of dimensionality, memory constraints, and limited training data. These challenges arise in many scientific and engineering applications, including subsurface flow, climate modeling, and geological carbon storage (GCS). In this work, we propose a scalable operator-learning framework based on the Karhunen-Loeve Deep Neural Network (KL-DNN) and demonstrate its performance for modeling GCS. The model is trained on a dataset comprising 100 samples of large-scale simulations in a three-dimensional domain with 1.7 million cells and 50 time steps. The KL-DNN method constructs latent spaces using low-rank singular value decomposition of static properties and a nested Karhunen-Loeve expansion for dynamic pressure fields, enabling full-resolution predictions without subsampling or spatial coarsening. The KL-DNN model achieves an average root mean square error (RMSE) of 1.1 psi for pressure (0.04% relative error with respect to the average pressure in the domain) and RMSE of 0.0146 for CO2 saturation (5% relative error with respect to the average saturation inside the plume). The model requires 20 minutes of training on a single GPU, representing a 19% reduction in the pressure errors, 7% reduction in the saturation error, and a two-order-of-magnitude speedup compared to DeepONet trained on the same dataset. These results, along with inference time of less than one minute, establish the proposed model as a practical and accurate solution for large-scale PDE problems, enabling rapid uncertainty quantification, history matching, and real-time decision support.
Latent neural operators improve the efficiency of operator learning for partial differential equations (PDEs) by performing the main computation on compact latent representations. However, directly compressing the input representation to obtain such compact representations may discard solution-relevant spatial information, especially for PDE solutions with multiscale structures. To address this problem, we propose HiLNO, a hierarchical latent neural operator that constructs a fine-to-coarse-to-fine latent space and further introduces multi-scale supervision (MSS) and anisotropic Gaussian attention. The hierarchy mitigates potential information loss during compression, while MSS aligns intermediate predictions with downsampled target fields, encouraging solution-relevant structures to be captured across multiple spatial scales. Anisotropic Gaussian attention enables feature transfer across the hierarchy, making HiLNO applicable to general geometries. Experiments on representative PDE benchmarks and a large-scale automotive aerodynamics task show that HiLNO achieves competitive predictive accuracy, while reducing the parameter count by an average of 84.4% and FLOPs by an average of 69.2% compared with LinearNO. Additional experiments demonstrate effective generalization to unseen spatial resolutions. Code is available at https://github.com/JcLimath/HiLNO.
This review examines neural operator architectures for learning solution operators of parametric partial differential equations (PDEs), with an emphasis on conceptual clarity and practical implementation. The work analyzes key models, including DeepONet, PCANet, and the Fourier Neural Operator, highlighting their underlying representations, computational structures, and comparative performance. These architectures are demonstrated on three canonical PDE problems: the Poisson equation, a linear elasticity problem, and a hyperelasticity problem. To make the presentation self-contained, key foundational topics are introduced, including finite-dimensional representations of function spaces, singular-value decomposition, and sampling from infinite-dimensional function spaces. Beyond forward modeling, the review discusses the use of neural operators as surrogate models within a Bayesian inverse-problem framework, including prior specification, forward-map approximation, and posterior computation. The performance of the three neural-operator architectures is evaluated on in-distribution samples, out-of-distribution samples, and Bayesian inference tasks. The review also discusses challenges related to prediction accuracy and generalization, outlining emerging strategies such as residual-based error correction and multi-level training. The review concludes by positioning neural operators within broader scientific-computing workflows and by identifying directions for reliable, scalable operator learning.
We present a natively multiscale operator learning method for the surrogate modeling of (numerical solvers for) multiscale partial differential equations (PDEs). The primary novelty of our method lies in a novel multiscale kernel frame function approximation technique. Leveraging this new kernel frame technique, we cast the operator learning problem as one of learning frame coefficients of output functions as a function of frame coefficients of input functions. The generalization step then automatically allows for a multiscale decomposition of the output functions. Our method is applicable to both tensor-product grids and point clouds. We present interpolation proofs, error estimates, and numerical convergence rates for our frame approximation. We the demonstrate the applicability of our method for the surrogate modeling of inherently multiscale PDEs. The new multiscale frame kernel method is significantly more accurate than popular neural operators on challenging problems from the literature, while simultaneously admitting an a posteriori multiscale decomposition upon generalization.
Branden Frieden, Ryan Whitehead, M. Keith Ballard +2