Disentangled Representation Learning for Parametric Partial Differential Equations

• URL: http://arxiv.org/abs/2410.02136v1
• Date: Thu, 3 Oct 2024 01:40:39 GMT
• Title: Disentangled Representation Learning for Parametric Partial Differential Equations
• Authors: Ning Liu, Lu Zhang, Tian Gao, Yue Yu,
• Abstract summary: We propose a new paradigm for learning disentangled representations from neural operator parameters. DisentangO is a novel hyper-neural operator architecture designed to unveil and disentangle the latent physical factors of variation embedded within the black-box neural operator parameters. We show that DisentangO effectively extracts meaningful and interpretable latent features, bridging the divide between predictive performance and physical understanding in neural operator frameworks.
• Score: 31.240283037552427
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Neural operators (NOs) have demonstrated remarkable success in learning mappings between function spaces, serving as efficient approximators for the forward solutions of complex physical systems governed by partial differential equations (PDEs). However, while effective as black-box solvers, they offer limited insight into the underlying physical mechanism, due to the lack of interpretable representations of the physical parameters that drive the system. To tackle this challenge, we propose a new paradigm for learning disentangled representations from neural operator parameters, thereby effectively solving an inverse problem. Specifically, we introduce DisentangO, a novel hyper-neural operator architecture designed to unveil and disentangle the latent physical factors of variation embedded within the black-box neural operator parameters. At the core of DisentangO is a multi-task neural operator architecture that distills the varying parameters of the governing PDE through a task-wise adaptive layer, coupled with a hierarchical variational autoencoder that disentangles these variations into identifiable latent factors. By learning these disentangled representations, our model not only enhances physical interpretability but also enables more robust generalization across diverse physical systems. Empirical evaluations across supervised, semi-supervised, and unsupervised learning contexts show that DisentangO effectively extracts meaningful and interpretable latent features, bridging the divide between predictive performance and physical understanding in neural operator frameworks.

Related papers

• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [63.5925701087252]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis. To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers. Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems [49.2319247825857]
  We show that diffusion-based generative models exhibit many properties favourable for neural operators. We propose to train a single model adaptable to multiple tasks, by alternating between the tasks during training.
  arXiv  Detail & Related papers  (2024-05-11T21:23:55Z)
• Physics-informed Discretization-independent Deep Compositional Operator Network [1.2430809884830318]
  We introduce a novel physics-informed model architecture which can generalize to various discrete representations of PDE parameters and irregular domain shapes. Inspired by deep operator neural networks, our model involves a discretization-independent learning of parameter embedding repeatedly. Numerical results demonstrate the accuracy and efficiency of the proposed method.
  arXiv  Detail & Related papers  (2024-04-21T12:41:30Z)
• Neural Parameter Regression for Explicit Representations of PDE Solution Operators [22.355460388065964]
  We introduce Neural Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs) NPR employs Physics-Informed Neural Network (PINN, Raissi et al., 2021) techniques to regress Neural Network (NN) parameters. The framework shows remarkable adaptability to new initial and boundary conditions, allowing for rapid fine-tuning and inference.
  arXiv  Detail & Related papers  (2024-03-19T14:30:56Z)
• Hierarchical Invariance for Robust and Interpretable Vision Tasks at Larger Scales [54.78115855552886]
  We show how to construct over-complete invariants with a Convolutional Neural Networks (CNN)-like hierarchical architecture. With the over-completeness, discriminative features w.r.t. the task can be adaptively formed in a Neural Architecture Search (NAS)-like manner. For robust and interpretable vision tasks at larger scales, hierarchical invariant representation can be considered as an effective alternative to traditional CNN and invariants.
  arXiv  Detail & Related papers  (2024-02-23T16:50:07Z)
• Manipulating Feature Visualizations with Gradient Slingshots [54.31109240020007]
  We introduce a novel method for manipulating Feature Visualization (FV) without significantly impacting the model's decision-making process. We evaluate the effectiveness of our method on several neural network models and demonstrate its capabilities to hide the functionality of arbitrarily chosen neurons.
  arXiv  Detail & Related papers  (2024-01-11T18:57:17Z)
• Deciphering and integrating invariants for neural operator learning with various physical mechanisms [22.508244510177683]
  We propose Physical Invariant Attention Neural Operator (PIANO) to decipher and integrate the physical invariants (PI) for operator learning from the PDE series with various physical mechanisms. Compared to existing techniques, PIANO can reduce the relative error by 13.6%-82.2% on PDE forecasting tasks across varying coefficients, forces, or boundary conditions.
  arXiv  Detail & Related papers  (2023-11-24T09:03:52Z)
• Interpretable Neural PDE Solvers using Symbolic Frameworks [0.0]
  Partial differential equations (PDEs) are ubiquitous in the world around us, modelling phenomena from heat and sound to quantum systems. Recent advances in deep learning have resulted in the development of powerful neural solvers. However, a significant challenge remains in their interpretability.
  arXiv  Detail & Related papers  (2023-10-31T13:56:25Z)
• Neural Operator: Learning Maps Between Function Spaces [75.93843876663128]
  We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations.
  arXiv  Detail & Related papers  (2021-08-19T03:56:49Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)

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.