# Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2210.12669 · HTML: https://mlsanity.com/paper/2210.12669/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 77 references that are in the map.
- First posted on arXiv 23 Oct 2022.

## Builds on (most cited first, 30 of 38)

- [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)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [RL2: Fast Reinforcement Learning via Slow Reinforcement Learning](https://mlsanity.com/paper/1611.02779/) (2016)
- [Introduction to Multi-Armed Bandits](https://mlsanity.com/paper/1904.07272/) (2019)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [TADAM: Task dependent adaptive metric for improved few-shot learning](https://mlsanity.com/paper/1805.10123/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Meta-SGD: Learning to Learn Quickly for Few-Shot Learning](https://mlsanity.com/paper/1707.09835/) (2017)
- [Learning to reinforcement learn](https://mlsanity.com/paper/1611.05763/) (2016)
- [Meta Networks](https://mlsanity.com/paper/1703.00837/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Probabilistic Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.02817/) (2018)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Learning to Optimize](https://mlsanity.com/paper/1606.01885/) (2016)
- [Bayesian Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.03836/) (2018)
- [Recasting Gradient-Based Meta-Learning as Hierarchical Bayes](https://mlsanity.com/paper/1801.08930/) (2018)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Infinite Mixture Prototypes for Few-Shot Learning](https://mlsanity.com/paper/1902.04552/) (2019)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)

## Built on by (most cited first, 6 of 6)

- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [A Sequential Meta-Transfer (SMT) Learning to Combat Complexities of Physics-Informed Neural Networks: Application to Composites Autoclave Processing](https://mlsanity.com/paper/2308.06447/) (2023)
- [Feature Mapping in Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2402.06955/) (2024)
- [Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metrics](https://mlsanity.com/paper/2402.11126/) (2024)
- [Tensor Gaussian Processes: Efficient Solvers for Nonlinear PDEs](https://mlsanity.com/paper/2510.13772/) (2025)
- [PINN Balls: Scaling Second-Order Methods for PINNs with Domain Decomposition and Adaptive Sampling](https://mlsanity.com/paper/2510.21262/) (2025)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) are emerging as popular mesh-free solvers for partial differential equations (PDEs). Recent extensions decompose the domain, apply different PINNs to solve the problem in each subdomain, and stitch the subdomains at the interface. Thereby, they can further alleviate the problem complexity, reduce the computational cost, and allow parallelization. However, the performance of multi-domain PINNs is sensitive to the choice of the interface conditions. While quite a few conditions have been proposed, there is no suggestion about how to select the conditions according to specific problems. To address this gap, we propose META Learning of Interface Conditions (METALIC), a simple, efficient yet powerful approach to dynamically determine appropriate interface conditions for solving a family of parametric PDEs. Specifically, we develop two contextual multi-arm bandit (MAB) models. The first one applies to the entire training course, and online updates a Gaussian process (GP) reward that given the PDE parameters and interface conditions predicts the performance. We prove a sub-linear regret bound for both UCB and Thompson sampling, which in theory guarantees the effectiveness of our MAB. The second one partitions the training into two stages, one is the stochastic phase and the other deterministic phase; we update a GP reward for each phase to enable different condition selections at the two stages to further bolster the flexibility and performance. We have shown the advantage of METALIC on four bench-mark PDE families.