Message Passing Neural PDE Solvers

Brandstetter, J.; Worrall, Daniel E.; Welling, Max

doi:10.48550/arxiv.2202.03376

Cited by 10 publications

(16 citation statements)

References 31 publications

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“…It maps initial conditions u 0 directly to solutions u at time t. The flavor we use predicts all N tout future times simultaneously, a technique called temporal bundling (Brandstetter et al, 2022). The initial conditions are also extended in time as a trajectory of the last N tin = 20 steps.…”

Section: Methodsmentioning

confidence: 99%

“…Again LPSDA improves both methods. This experiment is important because rollout stability and accuracy of autoregressive methods-a still poorly understood phenomenon-appears to be intimately connected with generalization capacity (Um et al, 2021;Brandstetter et al, 2022). LPSDA, in improving generalization performance, naturally improves AR methods as well as NO methods.…”

Section: Different Training Methodsmentioning

confidence: 99%

“…We used an AR-trained FNO model. We compare a naïve model predicting N tAR = 1 timestep at a time (∅), a temporally bundled model predicting N tAR = 120 timesteps at a time (TB), the N tAR = 120 model with the pushforward trick (Brandstetter et al, 2022) (TB+PF) needed for long run stability, and (TB+PF+LPDSA) where we add LPSDA. Figure 6 shows the resulting rollouts and Figure 7a shows the cumulative NMSE with respect to time.…”

Section: Very Long Rolloutsmentioning

confidence: 99%

“…In the field of neural PDE solvers, there are neural operator methods of finite dimensionality (Raissi, 2018;Sirignano & Spiliopoulos, 2018;Bhatnagar et al, 2019;Guo et al, 2016;Zhu & Zabaras, 2018;Khoo et al, 2020) and infinite dimensionality (Li et al, 2020;Bhattacharya et al, 2020;Patel et al, 2021). Leading autoregressive methods are Bar-Sinai et al (2019); Um et al (2021); Sanchez-Gonzalez et al (2020); Brandstetter et al (2022). Then in so-called neural augmentation a neural component is added to finite elements (Hsieh et al, 2019), multigrid solvers (Greenfeld et al, 2019), and eikonal solvers (Lichtenstein et al, 2019).…”

Section: Related Workmentioning

confidence: 99%

“…Whereas the former focus on processing spatial information, the latter process temporal information if temporal information is provided in the channel dimension. We therefore merge the temporal bundling idea of Brandstetter et al (2022) into the FNO framework, where K timesteps are input to the model. Consequently, the temporal information processing is strongly enhanced.…”

Section: Experimental Architecturesmentioning

confidence: 99%

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Lie Point Symmetry Data Augmentation for Neural PDE Solvers

Brandstetter¹,

Welling²,

Worrall³

2022

Preprint

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Neural networks are increasingly being used to solve partial differential equations (PDEs), replacing slower numerical solvers. However, a critical issue is that neural PDE solvers require highquality ground truth data, which usually must come from the very solvers they are designed to replace. Thus, we are presented with a proverbial chicken-and-egg problem. In this paper, we present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity-Lie point symmetry data augmentation (LPSDA). In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations, based on the Lie point symmetry group of the PDEs in question, something not possible in other application areas. We present this framework and demonstrate how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.

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Section: Methodsmentioning

confidence: 99%

Section: Different Training Methodsmentioning

confidence: 99%

Section: Very Long Rolloutsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Experimental Architecturesmentioning

confidence: 99%

See 3 more Smart Citations

Lie Point Symmetry Data Augmentation for Neural PDE Solvers

Brandstetter¹,

Welling²,

Worrall³

2022

Preprint

Self Cite

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AI-assisted modeling of capillary-driven droplet dynamics

Demou,

Savva

2023

DCE

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In this study, we present and assess data-driven approaches for modeling contact line dynamics, using droplet transport on chemically heterogeneous surfaces as a model system. Ground-truth data for training and validation are generated based on long-wave models that are applicable for slow droplet motion with small contact angles, which are known to accurately reproduce the dynamics with minimal computing resources compared to high-fidelity direct numerical simulations. The data-driven models are based on the Fourier neural operator (FNO) and are developed following two different approaches. The first deploys the data-driven method as an iterative neural network architecture, which predicts the future state of the contact line based on a number of previous states. The second approach corrects the time derivative of the contact line by augmenting its low-order asymptotic approximation with a data-driven counterpart, evolving the resulting system using standard time integration methods. The performance of each approach is evaluated in terms of accuracy and generalizability, concluding that the latter approach, although not originally explored within the original contribution on the FNO, outperforms the former.

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Plasma surrogate modelling using Fourier neural operators

Gopakumar,

Pamela,

Zanisi

et al. 2024

Nucl. Fusion

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Predicting plasma evolution within a Tokamak reactor is crucial to realizing the goal of sustainable fusion. Capabilities in forecasting the spatio-temporal evolution of plasma rapidly and accurately allow us to quickly iterate over design and control strategies on current Tokamak devices and future reactors. Modelling plasma evolution using numerical solvers is often expensive, consuming many hours on supercomputers, and hence, we need alternative inexpensive surrogate models. We demonstrate accurate predictions of plasma evolution both in simulation and experimental domains using deep learning-based surrogate modelling tools, viz., Fourier Neural Operators (FNO). We show that FNO has a speedup of six orders of magnitude over traditional solvers in predicting the plasma dynamics simulated from magnetohydrodynamic models, while maintaining a high accuracy (MSE ≈ 10−5). Our modified version of the FNO is capable of solving multi-variable Partial Differential Equations (PDE), and can capture the dependence among the different variables in a single model. FNOs can also predict plasma evolution on real-world experimental data observed by the cameras positioned within the MAST Tokamak, i.e., cameras looking across the central solenoid and the divertor in the Tokamak. We show that FNOs are able to accurately forecast the evolution of plasma and have the potential to be deployed for real-time monitoring. We also illustrate their capability in forecasting the plasma shape, the locations of interactions of the plasma with the central solenoid and the divertor for the full duration of the plasma shot within MAST. The FNO offers a viable alternative for surrogate modelling as it is quick to train and infer, and requires fewer data points, while being able to do zero-shot super-resolution and getting high-fidelity solutions.

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Message Passing Neural PDE Solvers

Cited by 10 publications

References 31 publications

Lie Point Symmetry Data Augmentation for Neural PDE Solvers

Lie Point Symmetry Data Augmentation for Neural PDE Solvers

AI-assisted modeling of capillary-driven droplet dynamics

Plasma surrogate modelling using Fourier neural operators

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