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

Lang, Quanjun; Lu, Fei

doi:10.48550/arxiv.2106.05565

Cited by 3 publications

(3 citation statements)

References 25 publications

(32 reference statements)

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“…Thus G is positive semi-definite. The operator L G is compact because G P L 2 pρ ˆρq, which follows from the fact that each u k is bounded and the definition of ρ (see also in [21]). Also, since G is positive semi-definite, so is L G .…”

Section: An Integral Operator and The Sida-rkhsmentioning

confidence: 97%

Data adaptive RKHS Tikhonov regularization for learning kernels in operators

Lu¹,

Lang²,

An³

2022

Preprint

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We present DARTR: a Data Adaptive RKHS Tikhonov Regularization method for the linear inverse problem of nonparametric learning of function parameters in operators. A key ingredient is a system intrinsic data adaptive (SIDA) RKHS, whose norm restricts the learning to take place in the function space of identifiability. DARTR utilizes this norm and selects the regularization parameter by the L-curve method. We illustrate its performance in examples including integral operators, nonlinear operators and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an accurate estimator robust to both numerical error due to discrete data and noise in data, and the estimator converges at a consistent rate as the data mesh refines under different levels of noises, outperforming two baseline regularizers using l 2 and L 2 norms.

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Section: An Integral Operator and The Sida-rkhsmentioning

confidence: 97%

Data adaptive RKHS Tikhonov regularization for learning kernels in operators

Lu¹,

Lang²,

An³

2022

Preprint

Self Cite

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“…These two works were done under a noiseless setting, i.e., the system evolves according to an ordinary differential equation and initial conditions of agents are i.i.d. As for the stochastic system, Li et al (2021) studied the learnability (identifiability) of interaction kernel by maximum likelihood estimator (MLE) under the coercivity condition, and Lang and Lu (2021) provided a complete characterization of learnability. Della Maestra and Hoffmann (2021) investigated a nonparametric estimation of the drift coefficient, and the interaction kernel can be separated by applying Fourier transform for deconvolution.…”

Section: Related Workmentioning

confidence: 99%

Mean-Field Nonparametric Estimation of Interacting Particle Systems

Yao¹,

Chen²,

Yang³

2022

Preprint

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This paper concerns the nonparametric estimation problem of the distribution-state dependent drift vector field in an interacting N -particle system. Observing single-trajectory data for each particle, we derive the mean-field rate of convergence for the maximum likelihood estimator (MLE), which depends on both Gaussian complexity and Rademacher complexity of the function class. In particular, when the function class contains α-smooth Hölder functions, our rate of convergence is minimax optimal on the order of N − α d+2α . Combining with a Fourier analytical deconvolution argument, we derive the consistency of MLE for the external force and interaction kernel in the McKean-Vlasov equation.

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“…Nonlocal operators: Nonlocal operators arise in various areas such as nonlocal and fractional diffusions [45,12,11,2,1,5,8,50,48], de-noising and regularization by nonlocal kernels [24,15,19], multi-agent systems with nonlocal interaction [37,36,26] and nonlocal networks [49,31]. The inverse problem for nonlocal diffusions has been studied in [22,28] from a single solution.…”

Section: Related Workmentioning

confidence: 99%

Nonparametric learning of kernels in nonlocal operators

Lu¹,

An²,

Yu³

2022

Preprint

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Nonlocal operators with integral kernels have become a popular tool for designing solution maps between function spaces, due to their efficiency in representing long-range dependence and the attractive feature of being resolution-invariant. In this work, we provide a rigorous identifiability analysis and convergence study for the learning of kernels in nonlocal operators. It is found that the kernel learning is an ill-posed or even ill-defined inverse problem, leading to divergent estimators in the presence of modeling errors or measurement noises. To resolve this issue, we propose a nonparametric regression algorithm with a novel data adaptive RKHS Tikhonov regularization method based on the function space of identifiability. The method yields a noisy-robust convergent estimator of the kernel as the data resolution refines, on both synthetic and real-world datasets. In particular, the method successfully learns a homogenized model for the stress wave propagation in a heterogeneous solid, revealing the unknown governing laws from real-world data at microscale. Our regularization method outperforms baseline methods in robustness, generalizability and accuracy.Preprint. Under review.

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

Cited by 3 publications

References 25 publications

Data adaptive RKHS Tikhonov regularization for learning kernels in operators

Data adaptive RKHS Tikhonov regularization for learning kernels in operators

Mean-Field Nonparametric Estimation of Interacting Particle Systems

Nonparametric learning of kernels in nonlocal operators

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