# Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs

@article{Wandel2021SplinePINNAP, title={Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs}, author={Nils Wandel and Michael Weinmann and Michael Neidlin and R. Klein}, journal={ArXiv}, year={2021}, volume={abs/2109.07143} }

Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed-form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the solution of PDEs based on a novel technique that combines the advantages of two recently emerging machine learning based approaches. First, physics-informed neural networks (PINNs) learn continuous solutions of PDEs and can be trained with little to no ground…

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## References

SHOWING 1-10 OF 48 REFERENCES

Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems

- Computer ScienceComputer Methods in Applied Mechanics and Engineering
- 2022

A novel discrete PINN framework based on graph convolutional network (GCN) and variational structure of PDE to solve forward and inverse partial differential equations (PDEs) in a unified manner and the use of a piecewise polynomial basis can reduce the dimension of search space and facilitate training and convergence.

SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels

- Computer Science2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition
- 2018

This work presents Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured and geometric input, e.g., graphs or meshes, that is a generalization of the traditional CNN convolution operator by using continuous kernel functions parametrized by a fixed number of trainable weights.

Train Once and Use Forever: Solving Boundary Value Problems in Unseen Domains with Pre-trained Deep Learning Models

- Computer ScienceArXiv
- 2021

A transferable framework for solving boundary value problems (BVPs) via deep neural networks which can be trained once and used forever for various domains of unseen sizes, shapes, and boundary conditions is introduced.

Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2020

This work proposes a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model non-linear dynamical systems without training data at a computational cost that is potentially magnitudes lower than standard numerical solvers.

DGM: A deep learning algorithm for solving partial differential equations

- Mathematics, EconomicsJournal of Computational Physics
- 2018

High-dimensional PDEs have been a longstanding computational challenge. We propose a deep learning algorithm similar in spirit to Galerkin methods, using a deep neural network instead of linear…

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

- Computer ScienceJ. Comput. Phys.
- 2019

Abstract We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear…

Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data

- Physics, Computer ScienceJ. Comput. Phys.
- 2019

This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions of the model predictive density and the reference conditional density as a minimization problem of the reverse Kullback-Leibler (KL) divergence.

Accelerating Eulerian Fluid Simulation With Convolutional Networks

- Computer ScienceICML
- 2017

This work proposes a data-driven approach that leverages the approximation power of deep-learning with the precision of standard solvers to obtain fast and highly realistic simulations of the Navier-Stokes equations.

Embedding Hard Physical Constraints in Neural Network Coarse-Graining of 3D Turbulence

- Computer Science, Physics
- 2020

A general framework to directly embed the notion of an incompressible fluid into Convolutional Neural Networks, and apply this to coarse-graining of turbulent flow on three-dimensional fully-developed turbulence is proposed.

Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification

- Physics, Computer ScienceJ. Comput. Phys.
- 2018

This approach achieves state of the art performance in terms of predictive accuracy and uncertainty quantification in comparison to other approaches in Bayesian neural networks as well as techniques that include Gaussian processes and ensemble methods even when the training data size is relatively small.