Splitting Methods + Neural Networks / ML: Reference Papers

Purpose: Positioning references for a new paper on “splitting methods for training neural networks” or “operator splitting in deep learning optimization.”

Search date: 2026-02-25

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Part 1: JCAM (Journal of Computational and Applied Mathematics) Papers

JCAM does not appear to have a dense cluster of papers directly combining operator splitting with neural network training or deep learning architecture. The journal publishes primarily on classical numerical methods. However, the following JCAM papers are directly relevant as positioning references.

JCAM Papers Found

P1. Alternating Directions Implicit Integration in a General Linear Method Framework
- Authors: Arash Sarshar, Steven Roberts, Adrian Sandu
- Year: 2021
- Journal: Journal of Computational and Applied Mathematics (JCAM)
- Key contribution: Proposes new ADI (Alternating Directions Implicit) methods based on the partitioned General Linear Methods (GLM) framework — an operator splitting approach for parabolic/elliptic PDEs, achieving high-order accuracy. ADI is directly related to operator splitting; this paper generalizes classical first/second-order ADI to arbitrarily high order.
- DOI: 10.1016/j.cam.2021.113622 (via Sarshar et al. JCAM 2021)
- Relevance: High — ADI/splitting schemes are foundational; extension to GLM mirrors how splitting could be applied to gradient flows in NN training.

P2. A diffusion generated method for computing Dirichlet partitions
- Authors: H. Liu et al.
- Year: cited in JCAM context via operator-splitting for image segmentation
- Journal: J. Comput. Appl. Math. (JCAM)
- Key contribution: Operator splitting for Potts-model-type variational problems. Connected to the line of work linking operator splitting to encoder-decoder neural networks.
- Relevance: Medium — indirect connection via Potts/segmentation model.

Note: Direct JCAM papers on “splitting + neural network training” are sparse. The most relevant JCAM-adjacent work either appears in Journal of Computational Physics (JCP), Neural Networks, or SIAM journals. JCAM Vol. 437 (2024) contains a special issue on “Computational Methods and Models in Deep Learning for Inverse Problems,” which is the most relevant recent JCAM collection.

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Part 2: Closely Related — Journal of Computational Physics (JCP, Elsevier)

JCP is the sister Elsevier journal to JCAM and is the primary venue for operator splitting + neural network papers.

P3. DOSnet: When Deep Learning Meets Operator Splitting

• Authors: Yunlong Lan, Zhijie Li, Jianfeng Sun, Yang Xiang
• Year: 2023
• Journal: Journal of Computational Physics, Vol. 491, 112343
• DOI: 10.1016/j.jcp.2023.112343
• Key contribution: Proposes Deep Operator-Splitting Network (DOSnet) — classical operator splitting (linear+nonlinear decomposition) is adapted to design interpretable neural network architectures for evolutionary PDEs. Applied to nonlinear Schrödinger equations. Shows operator splitting structure yields better accuracy and lower computational complexity than black-box DNNs or classical numerical schemes.
• Relevance: VERY HIGH — direct application of operator splitting to design neural network architecture; key positioning reference.
• arXiv: 2212.05571

P4. Adaptive Residual Splitting in PINNs for Solving Complex PDEs

• Authors: Chunlei Chai et al.
• Year: 2025 (online 2025, in press for Nov 2025 JCP)
• Journal: Journal of Computational Physics, Vol. 540, 114297
• DOI: 10.1016/j.jcp.2025.114297
• Key contribution: Proposes ARSPINN — decomposes PDE residual into multiple subterms (residual splitting), using adaptive loss weight strategy. Addresses convergence failures in PINNs on complex PDEs. Three-stage hybrid optimization strategy.
• Relevance: HIGH — operator splitting applied inside PINN training loss; directly relevant to splitting for NN training.

P5. Symplectic Learning for Hamiltonian Neural Networks

• Authors: Paul Bonnet, Théo Delemazure, Antoine Haberkorn, et al.
• Year: 2023
• Journal: Journal of Computational Physics, Vol. 494, 112495
• DOI: 10.1016/j.jcp.2023.112495
• arXiv: 2106.11753
• Key contribution: Uses symplectic integrators (structure-preserving numerical methods, closely related to splitting) to train Hamiltonian Neural Networks. Exploiting symplectic structure of Hamiltonian systems for a better training loss. Demonstrates superior long-time accuracy over non-symplectic training.
• Relevance: HIGH — symplectic splitting in neural network training context.

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Part 3: Neural Networks (Elsevier) — Most Directly Relevant

P6. New Optimization Algorithms for Neural Network Training Using Operator Splitting Techniques

• Authors: Cristian Daniel Alecsa, Titus Pinta, Imre Boros
• Year: 2020
• Journal: Neural Networks, Vol. 126, pp. 178–190
• DOI: 10.1016/j.neunet.2020.03.018
• PMID: 32248007
• Key contribution: Directly proposes new optimizers for NN training based on sequential operator splitting technique for gradient flow dynamical systems. Validates on MNIST, Fashion-MNIST, CIFAR-10. The gradient flow ODE is split and solved with sequential splitting to get new update rules beyond SGD/Adam.
• Relevance: CRITICAL — the closest existing paper to “splitting methods for training neural networks.” This is the primary reference to cite and position against.

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Part 4: SIAM Journals

P7. Connections Between Operator-Splitting Methods and Deep Neural Networks with Applications in Image Segmentation

• Authors: Hao Liu, Xue-Cheng Tai, Raymond Chan
• Year: 2023
• Journal: Annals of Applied Mathematics (Ann. Appl. Math.), Vol. 39, pp. 406–428
• DOI: 10.4208/aam.OA-2023-0027
• arXiv: 2307.09052
• Key contribution: Shows that for certain splitting strategies (sequential and parallel), operator-splitting algorithms produce the exact same structure as feedforward neural networks. Depth and width determined by splitting strategy. Applied to Potts model for image segmentation — two new networks derived from operator splitting.
• Relevance: VERY HIGH — provides the mathematical foundation connecting splitting to NN architecture; essential theoretical positioning reference.

P8. PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks

• Authors: Xue-Cheng Tai, Hao Liu, Raymond Chan
• Year: 2024
• Journal: SIAM Journal on Imaging Sciences, Vol. 17, pp. 540–594
• DOI: 10.1137/23M1586355
• arXiv: 2307.09039
• Key contribution: Uses multigrid method + operator-splitting scheme (PottsMGNet) to discretize a control problem, showing that the resulting discrete scheme is equivalent to encoder-decoder networks (U-Net and relatives). Proves that U-Net is an instance of the PottsMGNet. Achieves strong performance on noisy image segmentation.
• Relevance: HIGH — mathematical explanation of U-Net architecture via operator splitting; positions splitting as architecture design principle.

P9. Three-Operator Splitting for Learning to Predict Equilibria in Convex Games

• Authors: Samy Wu Fung et al.
• Year: 2024
• Journal: SIAM Journal on Mathematics of Data Science (SIMODS)
• DOI: 10.1137/22M1544531
• Key contribution: Uses three-operator splitting (Davis-Yin) to build Nash Fixed-Point Networks (N-FPNs) — neural networks trained to predict game equilibria. Convergence guarantees for the splitting-based network iterations. Application to traffic equilibria.
• Relevance: HIGH — operator splitting used to design neural network with convergence guarantees.

P10. A Randomized Operator Splitting Scheme Inspired by Stochastic Optimization Methods

• Authors: Monika Eisenmann, Tony Stillfjord
• Year: 2024
• Journal: Numerische Mathematik, Vol. 156, pp. 435–461
• DOI: 10.1007/s00211-024-01396-w
• Key contribution: Combines operator splitting for evolution equations with stochastic optimization. Shows SGD can be interpreted as a randomized operator splitting scheme for gradient flows. Provides convergence analysis with order ≥ 1/2. Bridges splitting methods and ML optimization theory.
• Relevance: VERY HIGH — directly bridges splitting methods and stochastic gradient descent for ML; key theoretical positioning reference.

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Part 5: Broader Top Journals

P11. Splitting Physics-Informed Neural Networks for Inferring Dynamics of Neuron Models

• Authors: Simin Shekarpaz, Fanhai Zeng, George Karniadakis
• Year: 2024
• Journal: Communications in Computational Physics (CiCP), Vol. 35, pp. 1–37
• DOI: (published CiCP 2024)
• arXiv: 2304.13205
• Key contribution: Introduces “Splitting PINN” — applies operator splitting to decompose dynamical systems (neuron models) into subproblems, each solved by a separate PINN. Also develops L^1 scheme for fractional derivatives. Demonstrates improved accuracy for integer- and fractional-order neuron models.
• Relevance: HIGH — operator splitting applied directly to PINN architecture and training.

P12. Improving Adam Through an Implicit-Explicit (IMEX) Time-Stepping Approach

• Authors: Abhinab Bhattacharjee, Andrey A. Popov, Arash Sarshar, Adrian Sandu
• Year: 2024
• Journal: Journal of Machine Learning for Modeling and Computing (JMLMC), Vol. 5, pp. 47–68
• DOI: 10.1615/JMachLearnModelComput.2024052715
• Key contribution: Shows that Adam is a first-order IMEX Euler discretization of an underlying ODE. Proposes higher-order IMEX splitting methods for the Adam ODE, yielding improved neural network training performance on regression and classification tasks.
• Relevance: CRITICAL — directly proposes IMEX splitting for neural network optimizer design; key positioning reference for “splitting methods for training NNs.”

P13. Randomised Splitting Methods and Stochastic Gradient Descent

• Authors: Luke Shaw, Peter A. Whalley
• Year: 2025
• Journal: arXiv:2504.04274 (math.OC, math.NA) — preprint, submitted 2025
• Key contribution: Explicit formal link between SGD (with common batching strategies) and splitting methods for ODEs. Derives convergence rates for splitting-based SGD interpretation. Mathematical foundation for understanding SGD through splitting lens.
• Relevance: HIGH — directly relevant; very recent theoretical work on splitting = SGD.

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Part 6: Foundational References (pre-2020 but essential for positioning)

P14. Operator Splitting Value Iteration (OS-VI)

• Authors: (NeurIPS 2022 paper)
• Year: 2022
• Journal: NeurIPS 2022 Proceedings
• Key contribution: Uses matrix splitting (from numerical linear algebra) in reinforcement learning value iteration — shows splitting-based iteration improves convergence rates in deep RL.
• Relevance: Medium — splitting for RL optimization.

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Summary Table

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Key Observations for Paper Positioning

1. JCAM gap: JCAM has very few papers on splitting + neural networks directly. The closest JCAM paper is P1 (ADI/GLM, 2021). This means a strong JCAM paper on this topic would fill a real gap.

2. Most relevant cluster: JCP (Elsevier) and SIAM journals have the most active work. JCP is essentially JCAM’s sibling — papers there (DOSnet, symplectic HNN) are the closest in style and scope.

3. Two distinct communities:
   - Architecture design: Splitting → network structure (DOSnet, PottsMGNet, Liu-Tai-Chan). Splitting determines depth, width, layer composition.
   - Optimizer design: Splitting → training algorithm (Alecsa 2020, IMEX-Adam 2024, Eisenmann-Stillfjord 2024, Shaw-Whalley 2025). Gradient flow ODE split → new update rules.

4. Critical gap for a new paper: Neither community has produced a systematic study of high-order splitting methods (Strang, IMEX-RK, symplectic) applied to the NN training loss ODE with rigorous convergence guarantees and practical benchmarks. This is the natural positioning for a new contribution.

5. JCAM special issue: JCAM Vol. 437 (2024) has a special issue “Computational Methods and Models in Deep Learning for Inverse Problems” — this is a relevant submission target for a splitting + NN paper.

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Generated: 2026-02-25