mlsanityData to 13 Oct 2026

Reduced-order modeling for parameterized PDEs via implicit neural representations

arXiv 2311.16410 ↗First posted 28 Nov 2023math.NA cs.LG cs.NAcited by 8

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Abstract (arXiv)

We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR), which models physics signals in a continuous manner and independent of spatial/temporal discretization. The proposed framework encodes PDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamics characterized by multiple PDE parameters. PNODE can be inferred by a hypernetwork to reduce the potential difficulties in learning PNODE due to a complex multilayer perceptron (MLP). The framework uses an INR to decode the latent dynamics and reconstruct accurate PDE solutions. Further, a physics-informed loss is also introduced to correct the prediction of unseen parameter instances. Incorporating the physics-informed loss also enables the model to be fine-tuned in an unsupervised manner on unseen PDE parameters. A numerical experiment is performed on a two-dimensional Burgers equation with a large variation of PDE parameters. We evaluate the proposed method at a large Reynolds number and obtain up to speedup of O(10^3) and ~1% relative error to the ground truth values.

Key facts. Cited by 8 papers in the map (data through 13 Oct 2026). Builds on 18 of its 38 references that are in the map. First posted on arXiv 28 Nov 2023.

Builds on

18 of 38 references.

References · 18

  1. 2018Neural Ordinary Differential Equationscited by 3,750
  2. 2020A fast and accurate physics-informed neural network reduced order model with shallow masked autoencodercited by 80
  3. 2022Continuous PDE Dynamics Forecasting with Implicit Neural Representationscited by 72
  4. 2022CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representationscited by 56
  5. 2020Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problemscited by 45
  6. 2022gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identificationcited by 30
  7. 2023GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencodercited by 30
  8. 2022Learning continuous models for continuous physicscited by 28
  9. 2021Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniquescited by 23
  10. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
  11. 2020Efficient nonlinear manifold reduced order modelcited by 17
  12. 2020Component-wise reduced order model lattice-type structure designcited by 16
6 more
  1. 2023Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systemscited by 14
  2. 2014Progressive construction of a parametric reduced-order model for PDE-constrained optimizationcited by 13
  3. 2022S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Modelscited by 8
  4. 2020A globally convergent method to accelerate topology optimization using on-the-fly model reductioncited by 4
  5. 2023Data-scarce surrogate modeling of shock-induced pore collapse processcited by 4
  6. 2022A globally convergent method to accelerate large-scale optimization using on-the-fly model hyperreduction: application to shape optimizationcited by 2

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Cited by year (2026 to 13 Oct)

Citing papers per year: 2024: 5; 2025: 2; 2026: 1.

8 papers.

  1. 2024On latent dynamics learning in nonlinear reduced order modelingcited by 14
  2. 2025CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operatorscited by 11
  3. 2024Advancing Generalization in PINNs through Latent-Space Representationscited by 5
  4. 2024Physics-informed reduced order model with conditional neural fieldscited by 4
  5. 2025A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEscited by 3
  6. 2024Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reductioncited by 2
  7. 2024Harnessing Loss Decomposition for Long-Horizon Wave Predictions via Deep Neural Networkscited by 2
  8. 2026Learning PDE solution operators with variable initial conditions via Latent Dynamics Networkscited by 0