Identifiability of interaction kernels in mean-field equations of interacting particles

• URL: http://arxiv.org/abs/2106.05565v4
• Date: Sat, 20 May 2023 16:02:19 GMT
• Title: Identifiability of interaction kernels in mean-field equations of interacting particles
• Authors: Quanjun Lang and Fei Lu
• Abstract summary: This study examines the identifiability of interaction kernels in mean-field equations of interacting particles or agents. We consider two data-adaptive $L2$ spaces: one weighted by a data-adaptive measure and the other using the Lebesgue measure. Our numerical demonstrations show that the weighted $L2$ space is preferable over the unweighted $L2$ space, as it yields more accurate regularized estimators.
• Score: 1.776746672434207
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: This study examines the identifiability of interaction kernels in mean-field equations of interacting particles or agents, an area of growing interest across various scientific and engineering fields. The main focus is identifying data-dependent function spaces where a quadratic loss functional possesses a unique minimizer. We consider two data-adaptive $L^2$ spaces: one weighted by a data-adaptive measure and the other using the Lebesgue measure. In each $L^2$ space, we show that the function space of identifiability is the closure of the RKHS associated with the integral operator of inversion. Alongside prior research, our study completes a full characterization of identifiability in interacting particle systems with either finite or infinite particles, highlighting critical differences between these two settings. Moreover, the identifiability analysis has important implications for computational practice. It shows that the inverse problem is ill-posed, necessitating regularization. Our numerical demonstrations show that the weighted $L^2$ space is preferable over the unweighted $L^2$ space, as it yields more accurate regularized estimators.

Related papers

• Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations [0.0]
  The appearance of singularities in the function of interest constitutes a fundamental challenge in scientific computing.<n>We propose a self-supervised learning framework for estimating the location of the singularity.<n>Various experiments are presented to demonstrate the ability of the proposed approach to deal with input perturbation, label corruption, and different kinds of singularities.
  arXiv  Detail & Related papers  (2025-06-29T17:39:41Z)
• Automatic feature selection and weighting in molecular systems using Differentiable Information Imbalance [41.452380773977154]
  Differentiable Information Imbalance (DII) is an automated method to rank information content between sets of features. Using distances in a ground truth feature space, DII identifies a low-dimensional subset of features that best preserves these relationships. DII can produce sparse solutions and determine the optimal size of the reduced feature space.
  arXiv  Detail & Related papers  (2024-10-30T11:19:10Z)
• D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operators [7.119066725173193]
  We propose a novel distributed approach to deal with input functions that exhibit heterogeneous properties. A central neural network is used to handle shared information across all output functions. We demonstrate that the corresponding neural network is a universal approximator of continuous nonlinear operators.
  arXiv  Detail & Related papers  (2023-10-29T03:29:59Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• Learning to Bound Counterfactual Inference in Structural Causal Models from Observational and Randomised Data [64.96984404868411]
  We derive a likelihood characterisation for the overall data that leads us to extend a previous EM-based algorithm. The new algorithm learns to approximate the (unidentifiability) region of model parameters from such mixed data sources. It delivers interval approximations to counterfactual results, which collapse to points in the identifiable case.
  arXiv  Detail & Related papers  (2022-12-06T12:42:11Z)
• Generative Adversarial Neural Operators [59.21759531471597]
  We propose the generative adversarial neural operator (GANO), a generative model paradigm for learning probabilities on infinite-dimensional function spaces. GANO consists of two main components, a generator neural operator and a discriminator neural functional. We empirically study GANOs in controlled cases where both input and output functions are samples from GRFs and compare its performance to the finite-dimensional counterpart GAN.
  arXiv  Detail & Related papers  (2022-05-06T05:12:22Z)
• Measuring dissimilarity with diffeomorphism invariance [94.02751799024684]
  We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces. We prove that DID enjoys properties which make it relevant for theoretical study and practical use.
  arXiv  Detail & Related papers  (2022-02-11T13:51:30Z)
• Multilevel orthogonal Bochner function subspaces with applications to robust machine learning [1.533771872970755]
  We consider the data as instances of a random field within a relevant Bochner space. Our key observation is that the classes can predominantly reside in two distinct subspaces.
  arXiv  Detail & Related papers  (2021-10-04T22:01:01Z)
• Data-driven discovery of interacting particle systems using Gaussian processes [3.0938904602244346]
  We study the data-driven discovery of distance-based interaction laws in second-order interacting particle systems. We propose a learning approach that models the latent interaction kernel functions as Gaussian processes. Numerical results on systems that exhibit different collective behaviors demonstrate efficient learning of our approach from scarce noisy trajectory data.
  arXiv  Detail & Related papers  (2021-06-04T22:00:53Z)
• Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles [1.776746672434207]
  We introduce a nonparametric algorithm to learn interaction kernels of mean-field equations for 1st-order systems of interacting particles. By at least squares with regularization, the algorithm learns the kernel on data-adaptive hypothesis spaces efficiently.
  arXiv  Detail & Related papers  (2020-10-29T15:37:17Z)
• The role of feature space in atomistic learning [62.997667081978825]
  Physically-inspired descriptors play a key role in the application of machine-learning techniques to atomistic simulations. We introduce a framework to compare different sets of descriptors, and different ways of transforming them by means of metrics and kernels. We compare representations built in terms of n-body correlations of the atom density, quantitatively assessing the information loss associated with the use of low-order features.
  arXiv  Detail & Related papers  (2020-09-06T14:12:09Z)
• Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories [13.3638879601361]
  We consider systems of interacting particles or agents, with dynamics determined by an interaction kernel. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator.
  arXiv  Detail & Related papers  (2020-07-30T01:28:06Z)

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.