Youngkyu Lee

Youngkyu Lee, Jongho Park, Chang-Ock Lee

The performance of neural networks has been significantly improved by increasing the number of channels in convolutional layers. However, this increase in performance comes with a higher computational cost, resulting in numerous studies focused on reducing it. One promising approach to address this issue is group convolution, which effectively reduces the computational cost by grouping channels. However, to the best of our knowledge, there has been no theoretical analysis on how well the group convolution approximates the standard convolution. In this paper, we mathematically analyze the approximation of the group convolution to the standard convolution with respect to the number of groups. Furthermore, we propose a novel variant of the group convolution called balanced group convolution, which shows a higher approximation with a small additional computational cost. We provide experimental results that validate our theoretical findings and demonstrate the superior performance of the balanced group convolution over other variants of group convolution.

Youngkyu Lee, Francesc Levrero Florencio, Jay Pathak et al.

The convergence behavior of classical iterative solvers for parametric partial differential equations (PDEs) is often highly sensitive to the domain and specific discretization of PDEs. Previously, we introduced hybrid solvers by combining the classical solvers with neural operators for a specific geometry 1, but they tend to under-perform in geometries not encountered during training. To address this challenge, we introduce Geo-DeepONet, a geometry-aware deep operator network that incorporates domain information extracted from finite element discretizations. Geo-DeepONet enables accurate operator learning across arbitrary unstructured meshes without requiring retraining. Building on this, we develop a class of geometry-aware hybrid preconditioned iterative solvers by coupling Geo-DeepONet with traditional methods such as relaxation schemes and Krylov subspace algorithms. Through numerical experiments on parametric PDEs posed over diverse unstructured domains, we demonstrate the enhanced robustness and efficiency of the proposed hybrid solvers for multiple real-world applications.

Youngkyu Lee, Shanqing Liu, Jerome Darbon et al.

We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a fixed-point neural operator (FPNO) that learns the direct mapping from the current iterate to the solution by emulating fixed-point iterations. Unlike traditional line-search or trust-region algorithms, the proposed FPNO adaptively employs negative step sizes to effectively mitigate the effects of unbalanced nonlinearities. Through numerical experiments we demonstrate the computational efficiency and robustness of the proposed NP-Newton method across multiple real-world applications, especially for very strong nonlinearities.

Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak et al.

The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural operators for specific geometries, but they tend to not perform well on geometries not previously seen during training. We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers, which, despite learning effectively across arbitrary unstructured meshes without requiring retraining, led to only modest reductions in iterations compared to state-of-the-art preconditioners. In this study we introduce Neural Subspace Proper Orthogonal Decomposition (NSPOD), a multigrid-like deep operator network-based preconditioner which can dramatically reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers, even when compared to state-of-the-art methods such as algebraic multigrid preconditioners. We demonstrate its efficiency via numerical experiments on a linearized version of solid mechanics PDEs applied to unstructured domains obtained from complex CAD geometries. We expect that the findings in this study lead to more efficient hybrid preconditioners that can match, or possibly even surpass, the convergence properties of the current gold standard preconditioning methods for solid mechanics PDEs.

Youngkyu Lee, Shanqing Liu, Zongren Zou et al.

Three-dimensional target identification using scattering techniques requires high accuracy solutions and very fast computations for real-time predictions in some critical applications. We first train a deep neural operator~(DeepONet) to solve wave propagation problems described by the Helmholtz equation in a domain \textit{without scatterers} but at different wavenumbers and with a complex absorbing boundary condition. We then design two classes of fast meta-solvers by combining DeepONet with either relaxation methods, such as Jacobi and Gauss-Seidel, or with Krylov methods, such as GMRES and BiCGStab, using the trunk basis of DeepONet as a coarse-scale preconditioner. We leverage the spectral bias of neural networks to account for the lower part of the spectrum in the error distribution while the upper part is handled inexpensively using relaxation methods or fine-scale preconditioners. The meta-solvers are then applied to solve scattering problems with different shape of scatterers, at no extra training cost. We first demonstrate that the resulting meta-solvers are shape-agnostic, fast, and robust, whereas the standard standalone solvers may even fail to converge without the DeepONet. We then apply both classes of meta-solvers to scattering from a submarine, a complex three-dimensional problem. We achieve very fast solutions, especially with the DeepONet-Krylov methods, which require orders of magnitude fewer iterations than any of the standalone solvers.

Alena Kopaničáková, Youngkyu Lee, George Em Karniadakis

We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient(PCG) method for solving parametric large-scale linear systems of equations. Unlike traditional deflation techniques that rely on eigenvector approximations or recycled Krylov subspaces, we generate the deflation subspaces using operator learning, specifically the Deep Operator Network~(DeepONet). To this aim, we introduce two complementary approaches for assembling the deflation operators. The first approach approximates near-null space vectors of the discrete PDE operator using the basis functions learned by the DeepONet. The second approach directly leverages solutions predicted by the DeepONet. To further enhance convergence, we also propose several strategies for prescribing the sparsity pattern of the deflation operator. A comprehensive set of numerical experiments encompassing steady-state, time-dependent, scalar, and vector-valued problems posed on both structured and unstructured geometries is presented and demonstrates the effectiveness of the proposed DeepONet-based deflated PCG method, as well as its generalization across a wide range of model parameters and problem resolutions.

Shilaj Baral, Youngkyu Lee, Sangam Khanal et al.

Purely data-driven surrogates for fluid dynamics often fail catastrophically from error accumulation, while existing hybrid methods have lacked the automation and robustness for practical use. To solve this, we developed XRePIT, a novel hybrid simulation strategy that synergizes machine learning (ML) acceleration with solver-based correction. We specifically designed our method to be fully automated and physics-aware, ensuring the stability and practical applicability that previous approaches lacked. We demonstrate that this new design overcomes long-standing barriers, achieving the first stable, accelerated rollouts for over 10,000 timesteps. The method also generalizes robustly to unseen boundary conditions and, crucially, scales to 3D flows. Our approach delivers speedups up to 4.98$\times$ while maintaining high physical fidelity, resolving thermal fields with relative errors of ~1E-3 and capturing low magnitude velocity dynamics with errors below 1E-2 ms-1. This work thus establishes a mature and scalable hybrid method, paving the way for its use in real-world engineering.

Sunwoong Yang, Youngkyu Lee, Namwoo Kang

This study presents an enhanced multi-fidelity Deep Operator Network (DeepONet) framework for efficient spatio-temporal flow field prediction when high-fidelity data is scarce. Key innovations include: a merge network replacing traditional dot-product operations, achieving 50.4% reduction in prediction error and 7.57% accuracy improvement while reducing training time by 96%; a transfer learning multi-fidelity approach that freezes pre-trained low-fidelity networks while making only the merge network trainable, outperforming alternatives by up to 76% and achieving 43.7% better accuracy than single-fidelity training; and a physics-guided subsampling method that strategically selects high-fidelity training points based on temporal dynamics, reducing high-fidelity sample requirements by 40% while maintaining comparable accuracy. Comprehensive experiments across multiple resolutions and datasets demonstrate the framework's ability to significantly reduce required high-fidelity dataset size while maintaining predictive accuracy, with consistent superior performance against conventional benchmarks.

Chang-Ock Lee, Youngkyu Lee, Byungeun Ryoo

Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which remain fixed throughout the computation, and uses a linear least squares method for training the parameters of the output layer of the neural network. It is known to be much faster than Physics informed neural networks. However, classical ELM is still computationally expensive when a high level of representation is desired in the solution as this requires solving a large least squares system. In this paper, we propose a nonoverlapping domain decomposition method (DDM) for ELMs that not only reduces the training time of ELMs, but is also suitable for parallel computation. In numerical analysis, DDMs have been widely studied to reduce the time to obtain finite element solutions for elliptic PDEs through parallel computation. Among these approaches, nonoverlapping DDMs are attracting the most attention. Motivated by these methods, we introduce local neural networks, which are valid only at corresponding subdomains, and an auxiliary variable at the interface. We construct a system on the variable and the parameters of local neural networks. A Schur complement system on the interface can be derived by eliminating the parameters of the output layer. The auxiliary variable is then directly obtained by solving the reduced system after which the parameters for each local neural network are solved in parallel. A method for initializing the hidden layer parameters suitable for high approximation quality in large systems is also proposed. Numerical results that verify the acceleration performance of the proposed method with respect to the number of subdomains are presented.

Youngkyu Lee, Alena Kopaničáková, George Em Karniadakis

We introduce a novel two-level overlapping additive Schwarz preconditioner for accelerating the training of scientific machine learning applications. The design of the proposed preconditioner is motivated by the nonlinear two-level overlapping additive Schwarz preconditioner. The neural network parameters are decomposed into groups (subdomains) with overlapping regions. In addition, the network's feed-forward structure is indirectly imposed through a novel subdomain-wise synchronization strategy and a coarse-level training step. Through a series of numerical experiments, which consider physics-informed neural networks and operator learning approaches, we demonstrate that the proposed two-level preconditioner significantly speeds up the convergence of the standard (LBFGS) optimizer while also yielding more accurate machine learning models. Moreover, the devised preconditioner is designed to take advantage of model-parallel computations, which can further reduce the training time.

Youngkyu Lee, Jongho Park, Chang-Ock Lee

Group convolution has been widely used in order to reduce the computation time of convolution, which takes most of the training time of convolutional neural networks. However, it is well known that a large number of groups significantly reduce the performance of group convolution. In this paper, we propose a new convolution methodology called ``two-level'' group convolution that is robust with respect to the increase of the number of groups and suitable for multi-GPU parallel computation. We first observe that the group convolution can be interpreted as a one-level block Jacobi approximation of the standard convolution, which is a popular notion in the field of numerical analysis. In numerical analysis, there have been numerous studies on the two-level method that introduces an intergroup structure that resolves the performance degradation issue without disturbing parallel computation. Motivated by these, we introduce a coarse-level structure which promotes intergroup communication without being a bottleneck in the group convolution. We show that all the additional work induced by the coarse-level structure can be efficiently processed in a distributed memory system. Numerical results that verify the robustness of the proposed method with respect to the number of groups are presented. Moreover, we compare the proposed method to various approaches for group convolution in order to highlight the superiority of the proposed method in terms of execution time, memory efficiency, and performance.

Chang-Ock Lee, Youngkyu Lee, Jongho Park

As deep neural networks (DNNs) become deeper, the training time increases. In this perspective, multi-GPU parallel computing has become a key tool in accelerating the training of DNNs. In this paper, we introduce a novel methodology to construct a parallel neural network that can utilize multiple GPUs simultaneously from a given DNN. We observe that layers of DNN can be interpreted as the time step of a time-dependent problem and can be parallelized by emulating a parallel-in-time algorithm called parareal. The parareal algorithm consists of fine structures which can be implemented in parallel and a coarse structure which gives suitable approximations to the fine structures. By emulating it, the layers of DNN are torn to form a parallel structure which is connected using a suitable coarse network. We report accelerated and accuracy-preserved results of the proposed methodology applied to VGG-16 and ResNet-1001 on several datasets.