mlsanityData to 14 Oct 2026

Evolutional Deep Neural Network

arXiv 2103.09959 ↗First posted 18 Mar 2021physics.comp-ph cs.LG physics.flu-dyncited by 50

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Abstract (arXiv)

The notion of an Evolutional Deep Neural Network (EDNN) is introduced for the solution of partial differential equations (PDE). The parameters of the network are trained to represent the initial state of the system only, and are subsequently updated dynamically, without any further training, to provide an accurate prediction of the evolution of the PDE system. In this framework, the network parameters are treated as functions with respect to the appropriate coordinate and are numerically updated using the governing equations. By marching the neural network weights in the parameter space, EDNN can predict state-space trajectories that are indefinitely long, which is difficult for other neural network approaches. Boundary conditions of the PDEs are treated as hard constraints, are embedded into the neural network, and are therefore exactly satisfied throughout the entire solution trajectory. Several applications including the heat equation, the advection equation, the Burgers equation, the Kuramoto Sivashinsky equation and the Navier-Stokes equations are solved to demonstrate the versatility and accuracy of EDNN. The application of EDNN to the incompressible Navier-Stokes equation embeds the divergence-free constraint into the network design so that the projection of the momentum equation to solenoidal space is implicitly achieved. The numerical results verify the accuracy of EDNN solutions relative to analytical and benchmark numerical solutions, both for the transient dynamics and statistics of the system.

Key facts. Cited by 50 papers in the map (data through 14 Oct 2026). Builds on 8 of its 32 references that are in the map. First posted on arXiv 18 Mar 2021.

Builds on

8 of 32 references.

References · 8

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  3. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 166
  4. 2020Deep Network Approximation for Smooth Functionscited by 164
  5. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 135
  6. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  7. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 118
  8. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 1; 2022: 5; 2023: 9; 2024: 11; 2025: 12; 2026: 12.

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  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  2. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  3. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 89
  4. 2024On conditional diffusion models for PDE simulationscited by 60
  5. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 56
  6. 2022Implicit Neural Spatial Representations for Time-dependent PDEscited by 35
  7. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  8. 2022Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustrationcited by 21
  9. 2023Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networkscited by 18
  10. 2023A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networkscited by 16
  11. 2023Energy-Dissipative Evolutionary Deep Operator Neural Networkscited by 14
  12. 2024CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equationscited by 14
38 more
  1. 2023Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemescited by 10
  2. 2023CoLA: Exploiting Compositional Structure for Automatic and Efficient Numerical Linear Algebracited by 7
  3. 2024Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equationscited by 7
  4. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
  5. 2024Neural networks for bifurcation and linear stability analysis of steady states in partial differential equationscited by 5
  6. 2024TENG: Time-Evolving Natural Gradient for Solving PDEs With Deep Neural Nets Toward Machine Precisioncited by 4
  7. 2023GRINN: A Physics-Informed Neural Network for solving hydrodynamic systems in the presence of self-gravitycited by 3
  8. 2023Physics-aware deep learning framework for linear elasticitycited by 2
  9. 2024A hybrid FEM-PINN method for time-dependent partial differential equationscited by 2
  10. 2024A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equationscited by 2
  11. 2025Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometriescited by 2
  12. 2024Multi evolutional deep neural networks (Multi-EDNN)cited by 1
  13. 2024Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domainscited by 1
  14. 2025Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force methodcited by 1
  15. 2025PMNO: A novel physics guided multi-step neural operator predictor for partial differential equationscited by 1
  16. 2025Low-Rank Evolutionary Deep Neural Networks via Adaptive Tangent-Space Reductioncited by 1
  17. 2025BEKAN: Boundary condition-guaranteed evolutionary Kolmogorov-Arnold networks with radial basis functions for solving PDE problemscited by 1
  18. 2025Control-Augmented Autoregressive Diffusion for Data Assimilationcited by 1
  19. 2025SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Lawscited by 1
  20. 2025Filtered Neural Galerkin model reduction schemes for efficient propagation of initial condition uncertainties in digital twinscited by 1
  21. 2026Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equationscited by 1
  22. 2026A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutionscited by 1
  23. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  24. 2024Data-driven 2D stationary quantum droplets and wave propagations in the amended GP equation with two potentials via deep neural networks learningcited by 0
  25. 2025Enhanced uncertainty quantification variational autoencoders for the solution of Bayesian inverse problemscited by 0
  26. 2025Maximum Principle of Optimal Probability Density Controlcited by 0
  27. 2025Solving PDEs With Deep Neural Nets under General Boundary Conditionscited by 0
  28. 2025MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equationscited by 0
  29. 2026TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditionscited by 0
  30. 2026Hybrid Quantum-Classical PINNs for Scientific Computing: A Multi-GPU Open-Source Frameworkcited by 0
  31. 2026Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solverscited by 0
  32. 2026Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problemscited by 0
  33. 2026A Novel Semantic Manifold Alignment Attack against Embedding-to-Embedding Obfuscation in Privacy-Preserving LLMscited by 0
  34. 2026Physics-enriched neural solvers for transient ice-flow simulationcited by 0
  35. 2026Domain-decomposed Evolutional Deep Neural Network with Random Features for Transient Pressure Diffusion with Discontinuous and High-Contrast Coefficientscited by 0
  36. 2026CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equationscited by 0
  37. 2026Amortized Score-Hamiltonian Policy Iteration: A Grid-Free Scheme for Relaxed Stochastic Control Problemscited by 0
  38. 2026CEENs: Causality-enforced evolutional networks for solving time-dependent partial differential equationscited by 0