From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven models

• URL: http://arxiv.org/abs/2606.08956v2
• Date: Tue, 09 Jun 2026 21:29:37 GMT
• Title: From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven models
• Abstract summary: We argue that many model types have a common structure, differing only in the assumed model class of the input-output relation they define.<n>Our analysis is intended to unite apparently disparate modeling strategies and provide insight into their appropriate use cases.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Scientists have historically relied on mathematical models based on differential equations to relate system inputs -- forces, fluxes, or heat sources -- to outputs, such as displacement, velocity, concentration, and temperature. These models rely on deep domain knowledge to determine the form of the governing differential equation, which is then calibrated with data by solving an inverse problem. In recent years, the field of Scientific Machine Learning has introduced a variety of alternative modeling strategies for physical systems. A method called Sparse Identification of Nonlinear Dynamics learns the governing equation as a sparse linear combination of terms in a user-defined library. Neural Ordinary Differential Equations construct the governing equation by taking in the state and its derivatives at the input layer of a neural network. Entirely foregoing the modeling framework of differential equations, neural operators directly learn a non-linear mapping between the system inputs and outputs. From inverse problems to neural operators, all of these modeling strategies can be conceptualized as data-driven machinery to predict a system's response over a range of inputs. It is then natural to wonder how exactly these various strategies relate to each other, and whether they can be neatly taxonomized. Drawing from the philosophical literature on scientific models, we argue that many model types have a common structure, differing only in the assumed model class of the input-output relation they define. Connecting to philosophical ideas on mechanism, and arguing that data from physical systems arises from solutions to parsimonious differential equations, we propose that only certain models are capable of mechanism discovery, and thus generalization. Our analysis is intended to unite apparently disparate modeling strategies and provide insight into their appropriate use cases.

Related papers

• No Equations Needed: Learning System Dynamics Without Relying on Closed-Form ODEs [56.78271181959529]
  This paper proposes a conceptual shift to modeling low-dimensional dynamical systems by departing from the traditional two-step modeling process.<n>Instead of first discovering a closed-form equation and then analyzing it, our approach, direct semantic modeling, predicts the semantic representation of the dynamical system.<n>Our approach not only simplifies the modeling pipeline but also enhances the transparency and flexibility of the resulting models.
  arXiv  Detail & Related papers  (2025-01-30T18:36:48Z)
• Physics-informed nonlinear vector autoregressive models for the prediction of dynamical systems [0.36248657646376703]
  We focus on one class of models called nonlinear vector autoregression (N VAR) to solve ordinary differential equations (ODEs) Motivated by connections to numerical integration and physics-informed neural networks, we explicitly derive the physics-informed N VAR. Because N VAR and piN VAR completely share their learned parameters, we propose an augmented procedure to jointly train the two models. We evaluate the ability of the piN VAR model to predict solutions to various ODE systems, such as the undamped spring, a Lotka-Volterra predator-prey nonlinear model, and the chaotic Lorenz system.
  arXiv  Detail & Related papers  (2024-07-25T14:10:42Z)
• Learning Governing Equations of Unobserved States in Dynamical Systems [0.0]
  We employ a hybrid neural ODE structure to learn governing equations of partially-observed dynamical systems. We demonstrate that the method is capable of successfully learning the true underlying governing equations of unobserved states within these systems.
  arXiv  Detail & Related papers  (2024-04-29T10:28:14Z)
• Learning Neural Differential Algebraic Equations via Operator Splitting [6.100037457394823]
  We present an Operator Splitting scheme for learning unknown components of DAEs from time-series data.<n>We show that the proposed OS-based time-stepping scheme is suitable for relevant system-theoretic data-driven modeling tasks.
  arXiv  Detail & Related papers  (2024-03-19T17:43:57Z)
• 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)
• Stretched and measured neural predictions of complex network dynamics [2.1024950052120417]
  Data-driven approximations of differential equations present a promising alternative to traditional methods for uncovering a model of dynamical systems. A recently employed machine learning tool for studying dynamics is neural networks, which can be used for data-driven solution finding or discovery of differential equations. We show that extending the model's generalizability beyond traditional statistical learning theory limits is feasible.
  arXiv  Detail & Related papers  (2023-01-12T09:44:59Z)
• Neural Laplace: Learning diverse classes of differential equations in the Laplace domain [86.52703093858631]
  We propose a unified framework for learning diverse classes of differential equations (DEs) including all the aforementioned ones. Instead of modelling the dynamics in the time domain, we model it in the Laplace domain, where the history-dependencies and discontinuities in time can be represented as summations of complex exponentials. In the experiments, Neural Laplace shows superior performance in modelling and extrapolating the trajectories of diverse classes of DEs.
  arXiv  Detail & Related papers  (2022-06-10T02:14:59Z)
• Capturing Actionable Dynamics with Structured Latent Ordinary Differential Equations [68.62843292346813]
  We propose a structured latent ODE model that captures system input variations within its latent representation. Building on a static variable specification, our model learns factors of variation for each input to the system, thus separating the effects of the system inputs in the latent space.
  arXiv  Detail & Related papers  (2022-02-25T20:00:56Z)
• On Neural Differential Equations [13.503274710499971]
  In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equations are two sides of the same coin. NDEs are suitable for tackling generative problems, dynamical systems, and time series. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides.
  arXiv  Detail & Related papers  (2022-02-04T23:32:29Z)
• Constructing Neural Network-Based Models for Simulating Dynamical Systems [59.0861954179401]
  Data-driven modeling is an alternative paradigm that seeks to learn an approximation of the dynamics of a system using observations of the true system. This paper provides a survey of the different ways to construct models of dynamical systems using neural networks. In addition to the basic overview, we review the related literature and outline the most significant challenges from numerical simulations that this modeling paradigm must overcome.
  arXiv  Detail & Related papers  (2021-11-02T10:51:42Z)
• Artificial neural network as a universal model of nonlinear dynamical systems [0.0]
  The map is built as an artificial neural network whose weights encode a modeled system. We consider the Lorenz system, the Roessler system and also Hindmarch-Rose neuron. High similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunovs exponents.
  arXiv  Detail & Related papers  (2021-03-06T16:02:41Z)

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.