Symbolic Regression for PDEs using Pruned Differentiable Programs

• URL: http://arxiv.org/abs/2303.07009v1
• Date: Mon, 13 Mar 2023 11:07:17 GMT
• Title: Symbolic Regression for PDEs using Pruned Differentiable Programs
• Authors: Ritam Majumdar, Vishal Jadhav, Anirudh Deodhar, Shirish Karande, Lovekesh Vig, Venkataramana Runkana
• Abstract summary: We introduce an end-to-end framework for obtaining mathematical expressions for solutions of Partial Differential Equations. We use a trained PINN to generate a dataset, upon which we perform Symbolic Regression. We observe a 95.3% reduction in parameters of DPA while maintaining accuracy at par with PINNs.
• Score: 12.23889788846524
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Physics-informed Neural Networks (PINNs) have been widely used to obtain accurate neural surrogates for a system of Partial Differential Equations (PDE). One of the major limitations of PINNs is that the neural solutions are challenging to interpret, and are often treated as black-box solvers. While Symbolic Regression (SR) has been studied extensively, very few works exist which generate analytical expressions to directly perform SR for a system of PDEs. In this work, we introduce an end-to-end framework for obtaining mathematical expressions for solutions of PDEs. We use a trained PINN to generate a dataset, upon which we perform SR. We use a Differentiable Program Architecture (DPA) defined using context-free grammar to describe the space of symbolic expressions. We improve the interpretability by pruning the DPA in a depth-first manner using the magnitude of weights as our heuristic. On average, we observe a 95.3% reduction in parameters of DPA while maintaining accuracy at par with PINNs. Furthermore, on an average, pruning improves the accuracy of DPA by 7.81% . We demonstrate our framework outperforms the existing state-of-the-art SR solvers on systems of complex PDEs like Navier-Stokes: Kovasznay flow and Taylor-Green Vortex flow. Furthermore, we produce analytical expressions for a complex industrial use-case of an Air-Preheater, without suffering from performance loss viz-a-viz PINNs.

Related papers

• Multimodal Policies with Physics-informed Representations [3.0253681238542076]
  In the control problems of the PDE systems, observation is important to make the decision. We propose a Physics-Informed Representation (PIR) algorithm for multimodal policies in PDE systems' control.
  arXiv  Detail & Related papers  (2024-10-20T01:56:15Z)
• Parameterized Physics-informed Neural Networks for Parameterized PDEs [24.926311700375948]
  In this paper, we propose a novel extension, parameterized physics-informed neural networks (PINNs) PINNs enable modeling the solutions of parameterized partial differential equations (PDEs) via explicitly encoding a latent representation of PDE parameters. We demonstrate that P$2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs.
  arXiv  Detail & Related papers  (2024-08-18T11:58:22Z)
• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• Unisolver: PDE-Conditional Transformers Are Universal PDE Solvers [55.0876373185983]
  We present the Universal PDE solver (Unisolver) capable of solving a wide scope of PDEs. Our key finding is that a PDE solution is fundamentally under the control of a series of PDE components. Unisolver achieves consistent state-of-the-art results on three challenging large-scale benchmarks.
  arXiv  Detail & Related papers  (2024-05-27T15:34:35Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Noise-aware Physics-informed Machine Learning for Robust PDE Discovery [5.746505534720594]
  This work is concerned with discovering the governing partial differential equation (PDE) of a physical system. Existing methods have demonstrated the PDE identification from finite observations but failed to maintain satisfying performance against noisy data. We introduce a noise-aware physics-informed machine learning framework to discover the governing PDE from data following arbitrary distributions.
  arXiv  Detail & Related papers  (2022-06-26T15:29:07Z)
• Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture [77.59766598165551]
  Physics-informed neural networks (PINNs) are revolutionizing science and engineering practice by bringing together the power of deep learning to bear on scientific computation. Here, we propose Auto-PINN, which employs Neural Architecture Search (NAS) techniques to PINN design. A comprehensive set of pre-experiments using standard PDE benchmarks allows us to probe the structure-performance relationship in PINNs.
  arXiv  Detail & Related papers  (2022-05-27T03:24:31Z)
• Lie Point Symmetry Data Augmentation for Neural PDE Solvers [69.72427135610106]
  We present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity. In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations. We show how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.
  arXiv  Detail & Related papers  (2022-02-15T18:43:17Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs [8.220908558735884]
  Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (NNs) to solve PDEs as a basis for data-driven inverse analysis. We propose the novel physics-informed convolutional-recurrent learning architectures (PhyCRNet and PhCRyNet-s) for solving PDEs without any labeled data.
  arXiv  Detail & Related papers  (2021-06-26T22:22:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.