Mathematical artificial data for operator learning

• URL: http://arxiv.org/abs/2507.06752v1
• Date: Wed, 09 Jul 2025 11:23:05 GMT
• Title: Mathematical artificial data for operator learning
• Authors: Heng Wu, Benzhuo Lu,
• Abstract summary: We present the Mathematical Artificial Data (MAD) framework, a new paradigm that integrates physical laws with data-driven learning to facilitate large-scale operator discovery.<n>We show MAD's generalizability and superior efficiency/accuracy across various differential equations scenarios.
• Score: 1.4579344926652846
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Machine learning has emerged as a transformative tool for solving differential equations (DEs), yet prevailing methodologies remain constrained by dual limitations: data-driven methods demand costly labeled datasets while model-driven techniques face efficiency-accuracy trade-offs. We present the Mathematical Artificial Data (MAD) framework, a new paradigm that integrates physical laws with data-driven learning to facilitate large-scale operator discovery. By exploiting DEs' intrinsic mathematical structure to generate physics-embedded analytical solutions and associated synthetic data, MAD fundamentally eliminates dependence on experimental or simulated training data. This enables computationally efficient operator learning across multi-parameter systems while maintaining mathematical rigor. Through numerical demonstrations spanning 2D parametric problems where both the boundary values and source term are functions, we showcase MAD's generalizability and superior efficiency/accuracy across various DE scenarios. This physics-embedded-data-driven framework and its capacity to handle complex parameter spaces gives it the potential to become a universal paradigm for physics-informed machine intelligence in scientific computing.

Related papers

• Geometric Operator Learning with Optimal Transport [77.16909146519227]
  We propose integrating optimal transport (OT) into operator learning for partial differential equations (PDEs) on complex geometries.<n>For 3D simulations focused on surfaces, our OT-based neural operator embeds the surface geometry into a 2D parameterized latent space.<n> Experiments with Reynolds-averaged Navier-Stokes equations (RANS) on the ShapeNet-Car and DrivAerNet-Car datasets show that our method achieves better accuracy and also reduces computational expenses.
  arXiv  Detail & Related papers  (2025-07-26T21:28:25Z)
• OmniFluids: Unified Physics Pre-trained Modeling of Fluid Dynamics [25.066485418709114]
  We introduce OmniFluids, a unified physics pre-trained operator learning framework.<n>It integrates physics-only pre-training, coarse-grid operator distillation, and few-shot fine-tuning.<n>It significantly outperforms state-of-the-art AI-driven methods in flow field reconstruction and turbulence statistics accuracy.
  arXiv  Detail & Related papers  (2025-06-12T16:23:02Z)
• High-fidelity Multiphysics Modelling for Rapid Predictions Using Physics-informed Parallel Neural Operator [17.85837423448985]
  Modelling complex multiphysics systems governed by nonlinear and strongly coupled partial differential equations (PDEs) is a cornerstone in computational science and engineering.<n>We propose a novel paradigm, physics-informed parallel neural operator (PIPNO), a scalable and unsupervised learning framework.<n>PIPNO efficiently captures nonlinear operator mappings across diverse physics, including geotechnical engineering, material science, electromagnetism, quantum mechanics, and fluid dynamics.
  arXiv  Detail & Related papers  (2025-02-26T20:29:41Z)
• Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning [3.912796219404492]
  SciML is a research field which combines physics-based and data-driven models.<n>Data-driven models aim to extract relations between input and output data.<n>We discuss the successful application of SciML to the simulation of the human cardiac function.
  arXiv  Detail & Related papers  (2025-01-30T19:09:38Z)
• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [60.58067866537143]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis.<n>To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers.<n> Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• Discovering physical laws with parallel combinatorial tree search [57.05912962368898]
  Symbolic regression plays a crucial role in scientific research thanks to its capability of discovering concise and interpretable mathematical expressions from data.<n>Existing algorithms have faced a critical bottleneck of accuracy and efficiency over a decade.<n>We introduce a parallel tree search (PCTS) model to efficiently distill generic mathematical expressions from limited data.
  arXiv  Detail & Related papers  (2024-07-05T10:41:15Z)
• Mechanistic Neural Networks for Scientific Machine Learning [58.99592521721158]
  We present Mechanistic Neural Networks, a neural network design for machine learning applications in the sciences. It incorporates a new Mechanistic Block in standard architectures to explicitly learn governing differential equations as representations. Central to our approach is a novel Relaxed Linear Programming solver (NeuRLP) inspired by a technique that reduces solving linear ODEs to solving linear programs.
  arXiv  Detail & Related papers  (2024-02-20T15:23:24Z)
• Discovering Interpretable Physical Models using Symbolic Regression and Discrete Exterior Calculus [55.2480439325792]
  We propose a framework that combines Symbolic Regression (SR) and Discrete Exterior Calculus (DEC) for the automated discovery of physical models. DEC provides building blocks for the discrete analogue of field theories, which are beyond the state-of-the-art applications of SR to physical problems. We prove the effectiveness of our methodology by re-discovering three models of Continuum Physics from synthetic experimental data.
  arXiv  Detail & Related papers  (2023-10-10T13:23:05Z)
• AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios [51.94807626839365]
  We propose the attention-inspired numerical solver (AttNS) to solve differential equations due to limited data.<n>AttNS is inspired by the effectiveness of attention modules in Residual Neural Networks (ResNet) in enhancing model generalization and robustness.
  arXiv  Detail & Related papers  (2023-02-05T01:39:21Z)
• Neural Operator: Is data all you need to model the world? An insight into the impact of Physics Informed Machine Learning [13.050410285352605]
  We provide an insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems. We highlight a novel and fast machine learning-based approach to learning the solution operator of a PDE operator learning.
  arXiv  Detail & Related papers  (2023-01-30T23:29:33Z)
• Advancing Reacting Flow Simulations with Data-Driven Models [50.9598607067535]
  Key to effective use of machine learning tools in multi-physics problems is to couple them to physical and computer models. The present chapter reviews some of the open opportunities for the application of data-driven reduced-order modeling of combustion systems.
  arXiv  Detail & Related papers  (2022-09-05T16:48:34Z)
• Physics-informed learning of governing equations from scarce data [14.95055620484844]
  This work introduces a physics-informed deep learning framework to discover governing partial differential equations (PDEs) from scarce and noisy representation data. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of PDE systems. The resulting computational framework shows the potential for closed-form model discovery in practical applications.
  arXiv  Detail & Related papers  (2020-05-05T22:13:22Z)

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.