Conformalized Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2405.08111v1
• Date: Mon, 13 May 2024 18:45:25 GMT
• Title: Conformalized Physics-Informed Neural Networks
• Authors: Lena Podina, Mahdi Torabi Rad, Mohammad Kohandel,
• Abstract summary: We introduce Conformalized PINNs (C-PINNs) to quantify the uncertainty of PINNs. C-PINNs utilize the framework of conformal prediction to quantify the uncertainty of PINNs.
• Score: 0.8437187555622164
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Physics-informed neural networks (PINNs) are an influential method of solving differential equations and estimating their parameters given data. However, since they make use of neural networks, they provide only a point estimate of differential equation parameters, as well as the solution at any given point, without any measure of uncertainty. Ensemble and Bayesian methods have been previously applied to quantify the uncertainty of PINNs, but these methods may require making strong assumptions on the data-generating process, and can be computationally expensive. Here, we introduce Conformalized PINNs (C-PINNs) that, without making any additional assumptions, utilize the framework of conformal prediction to quantify the uncertainty of PINNs by providing intervals that have finite-sample, distribution-free statistical validity.

Related papers

• RoPINN: Region Optimized Physics-Informed Neural Networks [66.38369833561039]
  Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) This paper proposes and theoretically studies a new training paradigm as region optimization. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm.
  arXiv  Detail & Related papers  (2024-05-23T09:45:57Z)
• Learning solutions of parametric Navier-Stokes with physics-informed neural networks [0.3989223013441816]
  We leverageformed-Informed Neural Networks (PINs) to learn solution functions of parametric Navier-Stokes equations (NSE) We consider the parameter(s) of interest as inputs of PINs along with coordinates, and train PINs on numerical solutions of parametric-PDES for instances of the parameters. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum.
  arXiv  Detail & Related papers  (2024-02-05T16:19:53Z)
• Correcting model misspecification in physics-informed neural networks (PINNs) [2.07180164747172]
  We present a general approach to correct the misspecified physical models in PINNs for discovering governing equations. We employ other deep neural networks (DNNs) to model the discrepancy between the imperfect models and the observational data. We envision that the proposed approach will extend the applications of PINNs for discovering governing equations in problems where the physico-chemical or biological processes are not well understood.
  arXiv  Detail & Related papers  (2023-10-16T19:25:52Z)
• Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence [65.63201894457404]
  We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of non-linear differential equations. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations.
  arXiv  Detail & Related papers  (2023-05-24T20:43:47Z)
• Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networks [2.569295887779268]
  Uncertainty Quantification (UQ) is just beginning to emerge in the context of PINNs. We propose a framework for UQ in Bayesian PINNs (B-PINNs) that incorporates the discrepancy between the B-PINN solution and the unknown true solution. We exploit recent results on error bounds for PINNs on linear dynamical systems and demonstrate the predictive uncertainty on a class of linear ODEs.
  arXiv  Detail & Related papers  (2022-12-14T01:15:26Z)
• Variational Neural Networks [88.24021148516319]
  We propose a method for uncertainty estimation in neural networks called Variational Neural Network (VNN) VNN generates parameters for the output distribution of a layer by transforming its inputs with learnable sub-layers. In uncertainty quality estimation experiments, we show that VNNs achieve better uncertainty quality than Monte Carlo Dropout or Bayes By Backpropagation methods.
  arXiv  Detail & Related papers  (2022-07-04T15:41:02Z)
• NUQ: Nonparametric Uncertainty Quantification for Deterministic Neural Networks [151.03112356092575]
  We show the principled way to measure the uncertainty of predictions for a classifier based on Nadaraya-Watson's nonparametric estimate of the conditional label distribution. We demonstrate the strong performance of the method in uncertainty estimation tasks on a variety of real-world image datasets.
  arXiv  Detail & Related papers  (2022-02-07T12:30:45Z)
• 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)
• Interval and fuzzy physics-informed neural networks for uncertain fields [0.0]
  Partial differential equations involving fuzzy and interval fields are traditionally solved using the finite element method. In this work we utilize physics-informed neural networks (PINNs) to solve interval and fuzzy partial differential equations. The resulting network structures termed interval physics-informed neural networks (iPINNs) and fuzzy physics-informed neural networks (fPINNs) show promising results.
  arXiv  Detail & Related papers  (2021-06-18T21:06:42Z)
• Frequentist Uncertainty in Recurrent Neural Networks via Blockwise Influence Functions [121.10450359856242]
  Recurrent neural networks (RNNs) are instrumental in modelling sequential and time-series data. Existing approaches for uncertainty quantification in RNNs are based predominantly on Bayesian methods. We develop a frequentist alternative that: (a) does not interfere with model training or compromise its accuracy, (b) applies to any RNN architecture, and (c) provides theoretical coverage guarantees on the estimated uncertainty intervals.
  arXiv  Detail & Related papers  (2020-06-20T22:45:32Z)
• Optimal Uncertainty-guided Neural Network Training [14.768115786212187]
  We propose a highly customizable smooth cost function for developing NNs to construct optimal PIs. Results show that the proposed method reduces variation in the quality of PIs, accelerates the training, and improves convergence probability from 99.2% to 99.8%.
  arXiv  Detail & Related papers  (2019-12-30T00:03:28Z)

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.