# Meta-Auto-Decoder for Solving Parametric Partial Differential Equations

arXiv: https://arxiv.org/abs/2111.08823 · HTML: https://mlsanity.com/paper/2111.08823/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 46 references that are in the map.
- First posted on arXiv 15 Nov 2021.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [How to train your MAML](https://mlsanity.com/paper/1810.09502/) (2018)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [MetaSDF: Meta-learning Signed Distance Functions](https://mlsanity.com/paper/2006.09662/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Bayesian Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.03836/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [MeshfreeFlowNet: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework](https://mlsanity.com/paper/2005.01463/) (2020)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [Solving Partial Differential Equations with Point Source Based on Physics-Informed Neural Networks](https://mlsanity.com/paper/2111.01394/) (2021)

## Built on by (most cited first, 30 of 36)

- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Minimax Optimal Kernel Operator Learning via Multilevel Training](https://mlsanity.com/paper/2209.14430/) (2022)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods](https://mlsanity.com/paper/2410.06820/) (2024)
- [OmniArch: Building Foundation Model For Scientific Computing](https://mlsanity.com/paper/2402.16014/) (2024)
- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)
- [GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning](https://mlsanity.com/paper/2410.23889/) (2024)
- [Principled Acceleration of Iterative Numerical Methods Using Machine Learning](https://mlsanity.com/paper/2206.08594/) (2022)
- [LatentPINNs: Generative physics-informed neural networks via a latent representation learning](https://mlsanity.com/paper/2305.07671/) (2023)
- [Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs](https://mlsanity.com/paper/2310.13270/) (2023)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Advancing Generalizability by the Baldwin Effect](https://mlsanity.com/paper/2312.03243/) (2023)
- [Spatio-Temporal Fluid Dynamics Modeling via Physical-Awareness and Parameter Diffusion Guidance](https://mlsanity.com/paper/2403.13850/) (2024)
- [Advancing Generalization in PINNs through Latent-Space Representations](https://mlsanity.com/paper/2411.19125/) (2024)
- [Adaptive Physics-informed Neural Networks: A Survey](https://mlsanity.com/paper/2503.18181/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference](https://mlsanity.com/paper/2602.01493/) (2026)
- [Residuals Are Not Enough: Limits of Physics-Informed Pre-Training for Scientific Foundation Models](https://mlsanity.com/paper/2503.19081/) (2025)
- [Ricci flow regularization in latent spaces for the forward learning of partial differential equations](https://mlsanity.com/paper/2401.14591/) (2024)
- [Towards Automated Knowledge Integration From Human-Interpretable Representations](https://mlsanity.com/paper/2402.16105/) (2024)
- [Base Models for Parabolic Partial Differential Equations](https://mlsanity.com/paper/2407.12234/) (2024)
- [S2GPT-PINNs: Sparse and Small models for PDEs](https://mlsanity.com/paper/2506.15687/) (2025)
- [Meta-Learning for Airflow Simulations with Graph Neural Networks](https://mlsanity.com/paper/2306.10624/) (2023)
- [GIT-Net: Generalized Integral Transform for Operator Learning](https://mlsanity.com/paper/2312.02450/) (2023)
- [Neumann Series-based Neural Operator for Solving Inverse Medium Problem](https://mlsanity.com/paper/2409.09480/) (2024)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operators](https://mlsanity.com/paper/2506.19396/) (2025)
- [Physics-Informed Chebyshev Polynomial Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2602.01737/) (2026)

## Abstract (arXiv)

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc. Recently, building learning-based numerical solvers for parametric PDEs has become an emerging new field. One category of methods such as the Deep Galerkin Method (DGM) and Physics-Informed Neural Networks (PINNs) aim to approximate the solution of the PDEs. They are typically unsupervised and mesh-free, but require going through the time-consuming network training process from scratch for each set of parameters of the PDE. Another category of methods such as Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) try to approximate the solution mapping directly. Being fast with only one forward inference for each PDE parameter without retraining, they often require a large corpus of paired input-output observations drawn from numerical simulations, and most of them need a predefined mesh as well. In this paper, we propose Meta-Auto-Decoder (MAD), a mesh-free and unsupervised deep learning method that enables the pre-trained model to be quickly adapted to equation instances by implicitly encoding (possibly heterogenous) PDE parameters as latent vectors. The proposed method MAD can be interpreted by manifold learning in infinite-dimensional spaces, granting it a geometric insight. Extensive numerical experiments show that the MAD method exhibits faster convergence speed without losing accuracy than other deep learning-based methods. The project page with code is available: https://gitee.com/mindspore/mindscience/tree/master/MindElec/.