mlsanityData to 14 Oct 2026

RiemannONets: Interpretable Neural Operators for Riemann Problems

arXiv 2401.08886 ↗First posted 16 Jan 2024cs.LG physics.flu-dyncited by 28

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Abstract (arXiv)

Developing the proper representations for simulating high-speed flows with strong shock waves, rarefactions, and contact discontinuities has been a long-standing question in numerical analysis. Herein, we employ neural operators to solve Riemann problems encountered in compressible flows for extreme pressure jumps (up to 1010 pressure ratio). In particular, we first consider the DeepONet that we train in a two-stage process, following the recent work of citelee2023training, wherein the first stage, a basis is extracted from the trunk net, which is orthonormalized and subsequently is used in the second stage in training the branch net. This simple modification of DeepONet has a profound effect on its accuracy, efficiency, and robustness and leads to very accurate solutions to Riemann problems compared to the vanilla version. It also enables us to interpret the results physically as the hierarchical data-driven produced basis reflects all the flow features that would otherwise be introduced using ad hoc feature expansion layers. We also compare the results with another neural operator based on the U-Net for low, intermediate, and very high-pressure ratios that are very accurate for Riemann problems, especially for large pressure ratios, due to their multiscale nature but computationally more expensive. Overall, our study demonstrates that simple neural network architectures, if properly pre-trained, can achieve very accurate solutions of Riemann problems for real-time forecasting. The source code, along with its corresponding data, can be found at the following URL: https://github.com/apey236/RiemannONet/tree/main

Key facts. Cited by 28 papers in the map (data through 14 Oct 2026). Builds on 23 of its 44 references that are in the map. First posted on arXiv 16 Jan 2024.

Builds on

23 of 44 references.

References · 23

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  2. 2016Gaussian Error Linear Units (GELUs)cited by 3,502
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  5. 2018Group Normalizationcited by 1,524
  6. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  7. 2022U-NO: U-shaped Neural Operatorscited by 230
  8. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 197
  9. 2016A guide to convolution arithmetic for deep learningcited by 185
  10. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  11. 2022Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transportcited by 92
  12. 2022Wavelet neural operator: a neural operator for parametric partial differential equationscited by 78
11 more
  1. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 76
  2. 2022Learning two-phase microstructure evolution using neural operators and autoencoder architecturescited by 72
  3. 2021Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functionscited by 49
  4. 2022Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priorscited by 39
  5. 2022SVD Perspectives for Augmenting DeepONet Flexibility and Interpretabilitycited by 35
  6. 2023On the training and generalization of deep operator networkscited by 31
  7. 2022DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectoriescited by 30
  8. 2023Learning stiff chemical kinetics using extended deep neural operatorscited by 30
  9. 2022How important are activation functions in regression and classification? A survey, performance comparison, and future directionscited by 29
  10. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  11. 2023Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO)cited by 16

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 12; 2025: 10; 2026: 6.

28 papers.

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  1. 2024Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modelingcited by 36
  2. 2024Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularitycited by 34
  3. 2025Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systemscited by 31
  4. 2024Basis-to-Basis Operator Learning Using Function Encoderscited by 17
  5. 2025Fusion-DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic and Supersonic Flowscited by 13
  6. 2024Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architecturescited by 12
  7. 2024Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learningcited by 11
  8. 2025Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEscited by 11
  9. 2024Ensemble and Mixture-of-Experts DeepONets For Operator Learningcited by 9
  10. 2024GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Lawscited by 8
  11. 2025ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithmscited by 7
  12. 2024A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problemscited by 6
16 more
  1. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  2. 2025BubbleOKAN: A Physics-Informed Interpretable Neural Operator for High-Frequency Bubble Dynamicscited by 5
  3. 2026FEKAN: Feature-Enriched Kolmogorov-Arnold Networkscited by 5
  4. 2025Probabilistic Predictions of Process-Induced Deformation in Carbon/Epoxy Composites Using a Deep Operator Networkcited by 3
  5. 2024Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reductioncited by 2
  6. 2025Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learningcited by 2
  7. 2025AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kineticscited by 2
  8. 2024Fourier neural operators for spatiotemporal dynamics in two-dimensional turbulencecited by 1
  9. 2026In Transformer We Trust? A Perspective on Transformer Architecture Failure Modescited by 1
  10. 2026GenVoid: Uncertainty-Aware Learning of Subsurface Material Defects with an Experimentally Validated Physics-Informed Generative Modelcited by 1
  11. 2024A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical resultscited by 0
  12. 2024Variational autoencoders with latent high-dimensional steady geometric flows for dynamicscited by 0
  13. 2025Crack Path Prediction with Operator Learning using Discrete Particle System data Generationcited by 0
  14. 2026Learning Contractive Integral Operators with Fredholm Integral Neural Operatorscited by 0
  15. 2026A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Lawscited by 0
  16. 2026Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fieldscited by 0