mlsanityData to 7 Oct 2026

A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

arXiv 2103.06183 ↗First posted 10 Mar 2021math.NA cs.LG cs.NAcited by 28

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Abstract (arXiv)

Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n-width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. To provide guidelines for the design of deep autoencoders, we consider a nonlinear version of the Kolmogorov n-width over which we base the concept of a minimal latent dimension. We show that the latter is intimately related to the topological properties of the solution manifold, and we provide theoretical results with particular emphasis on second order elliptic PDEs, characterizing the minimal dimension and the approximation errors of the proposed approach. The theory presented is further supported by numerical experiments, where we compare the proposed approach with classical POD-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). Builds on 20 of its 75 references that are in the map. First posted on arXiv 10 Mar 2021.

Builds on

20 of 75 references.

References · 20

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8 more
  1. 2017Optimal Approximation with Sparsely Connected Deep Neural Networkscited by 90
  2. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
  3. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  4. 2019Approximation Capabilities of Neural ODEs and Invertible Residual Networkscited by 56
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 4; 2023: 3; 2024: 8; 2025: 5; 2026: 8.

28 papers.

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  1. 2022Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutionscited by 29
  2. 2025Reduced Order Modeling with Shallow Recurrent Decoder Networkscited by 24
  3. 2023Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networkscited by 17
  4. 2024On latent dynamics learning in nonlinear reduced order modelingcited by 14
  5. 2024Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learningcited by 12
  6. 2022Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order modelscited by 10
  7. 2023Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemescited by 10
  8. 2024A practical existence theorem for reduced order models based on convolutional autoencoderscited by 9
  9. 2022Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEscited by 7
  10. 2022Non-linear manifold ROM with Convolutional Autoencoders and Reduced Over-Collocation methodcited by 7
  11. 2024TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNscited by 5
  12. 2025Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimationcited by 5
16 more
  1. 2024PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEscited by 4
  2. 2024Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order modelscited by 4
  3. 2025Deep Symmetric Autoencoders from the Eckart-Young-Schmidt Perspectivecited by 2
  4. 2023A Note on Dimensionality Reduction in Deep Neural Networks using Empirical Interpolation Methodcited by 1
  5. 2026Nonlinear model reduction for transport-dominated problemscited by 1
  6. 2026Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagatorscited by 1
  7. 2024An Efficient Deep Learning Approach for Approximating Parameter-to-Solution Maps of PDEscited by 0
  8. 2024Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order modelscited by 0
  9. 2025Data-Driven Model Reduction using WeldNet: Windowed Encoders for Learning Dynamicscited by 0
  10. 2025Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inferencecited by 0
  11. 2026An adjoint method for training data-driven reduced-order modelscited by 0
  12. 2026Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEscited by 0
  13. 2026AE-ViT: Stable Long-Horizon Parametric Partial Differential Equations Modelingcited by 0
  14. 2026Two-Valued Symmetric Circulant Matrices: Applications in Deep Learningcited by 0
  15. 2026Conformal risk control for model-form uncertainty in parametric non-intrusive reduced-order modelscited by 0
  16. 2026Variational Parameter Calibration with Physics-Aware Latent-Space Surrogatescited by 0