mlsanityData to 14 Oct 2026

Energy-Dissipative Evolutionary Deep Operator Neural Networks

arXiv 2306.06281 ↗First posted 9 Jun 2023stat.ML cs.LGcited by 14

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Abstract (arXiv)

Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seed numerical solutions for a class of partial differential equations instead of a single partial differential equation, such as partial differential equations with different parameters or different initial conditions. The network consists of two sub-networks, the Branch net and the Trunk net. For an objective operator G, the Branch net encodes different input functions u at the same number of sensors, and the Trunk net evaluates the output function at any location. By minimizing the error between the evaluated output q and the expected output G(u)(y), DeepONet generates a good approximation of the operator G. In order to preserve essential physical properties of PDEs, such as the Energy Dissipation Law, we adopt a scalar auxiliary variable approach to generate the minimization problem. It introduces a modified energy and enables unconditional energy dissipation law at the discrete level. By taking the parameter as a function of time t, this network can predict the accurate solution at any further time with feeding data only at the initial state. The data needed can be generated by the initial conditions, which are readily available. In order to validate the accuracy and efficiency of our neural networks, we provide numerical simulations of several partial differential equations, including heat equations, parametric heat equations and Allen-Cahn equations.

Key facts. Cited by 14 papers in the map (data through 14 Oct 2026). Builds on 15 of its 36 references that are in the map. First posted on arXiv 9 Jun 2023.

Builds on

15 of 36 references.

References · 15

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  5. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  6. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
  7. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
  8. 2019Neural Jump Stochastic Differential Equationscited by 150
  9. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
  10. 2018Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systemscited by 89
  11. 2020A physics-informed operator regression framework for extracting data-driven continuum modelscited by 60
  12. 2021Evolutional Deep Neural Networkcited by 50
3 more
  1. 2019Data-Driven Deep Learning of Partial Differential Equations in Modal Spacecited by 46
  2. 2020Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEscited by 40
  3. 2019Learning functionals via LSTM neural networks for predicting vessel dynamics in extreme sea statescited by 12

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 5; 2025: 6; 2026: 3.

14 papers.

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  1. 2024Basis-to-Basis Operator Learning Using Function Encoderscited by 17
  2. 2024Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learningcited by 15
  3. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  4. 2024A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problemscited by 6
  5. 2026Muon with Spectral Guidance: Efficient Optimization for Scientific Machine Learningcited by 4
  6. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  7. 2025Learn Singularly Perturbed Solutions via Homotopy Dynamicscited by 2
  8. 2025Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structurescited by 1
  9. 2025Low-Rank Evolutionary Deep Neural Networks via Adaptive Tangent-Space Reductioncited by 1
  10. 2025BEKAN: Boundary condition-guaranteed evolutionary Kolmogorov-Arnold networks with radial basis functions for solving PDE problemscited by 1
  11. 2026Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equationscited by 1
  12. 2024An Energy-Based Self-Adaptive Learning Rate for Stochastic Gradient Descent: Enhancing Unconstrained Optimization with VAV methodcited by 0
2 more
  1. 2025Accelerating Particle-based Energetic Variational Inferencecited by 0
  2. 2026DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splittingcited by 0