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RIGOROUS JUSTIFICATION OF THE FOKKER-PLANCK EQUATIONS OF NEURAL NETWORKS BASED ON AN ITERATION PERSPECTIVE

Publication ,  Journal Article
Liu, JG; Wang, Z; Zhang, Y; Zhou, Z
Published in: SIAM Journal on Mathematical Analysis
January 1, 2022

In this work, the primary goal is to establish a rigorous connection between the Fokker-Planck equation of neural networks and its microscopic model: the diffusion-jump stochastic process that captures the mean-field behavior of collections of neurons in the integrate-and-fire model. The proof is based on a novel iteration scheme: with an auxiliary random variable counting the firing events, both the density function of the stochastic process and the solution of the PDE problem admit series representations, and thus the difficulty in verifying the link between the density function and the PDE solution in each subproblem is greatly mitigated. The iteration approach provides a generic framework for integrating the probability approach with PDE techniques, with which we prove that the density function of the diffusion-jump stochastic process is indeed the classical solution of the Fokker-Planck equation with a unique flux-shift structure.

Duke Scholars

Published In

SIAM Journal on Mathematical Analysis

DOI

EISSN

1095-7154

ISSN

0036-1410

Publication Date

January 1, 2022

Volume

54

Issue

1

Start / End Page

1270 / 1312

Related Subject Headings

  • Applied Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

Citation

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MLA
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Liu, J. G., Wang, Z., Zhang, Y., & Zhou, Z. (2022). RIGOROUS JUSTIFICATION OF THE FOKKER-PLANCK EQUATIONS OF NEURAL NETWORKS BASED ON AN ITERATION PERSPECTIVE. SIAM Journal on Mathematical Analysis, 54(1), 1270–1312. https://doi.org/10.1137/20M1338368
Liu, J. G., Z. Wang, Y. Zhang, and Z. Zhou. “RIGOROUS JUSTIFICATION OF THE FOKKER-PLANCK EQUATIONS OF NEURAL NETWORKS BASED ON AN ITERATION PERSPECTIVE.” SIAM Journal on Mathematical Analysis 54, no. 1 (January 1, 2022): 1270–1312. https://doi.org/10.1137/20M1338368.
Liu JG, Wang Z, Zhang Y, Zhou Z. RIGOROUS JUSTIFICATION OF THE FOKKER-PLANCK EQUATIONS OF NEURAL NETWORKS BASED ON AN ITERATION PERSPECTIVE. SIAM Journal on Mathematical Analysis. 2022 Jan 1;54(1):1270–312.
Liu, J. G., et al. “RIGOROUS JUSTIFICATION OF THE FOKKER-PLANCK EQUATIONS OF NEURAL NETWORKS BASED ON AN ITERATION PERSPECTIVE.” SIAM Journal on Mathematical Analysis, vol. 54, no. 1, Jan. 2022, pp. 1270–312. Scopus, doi:10.1137/20M1338368.
Liu JG, Wang Z, Zhang Y, Zhou Z. RIGOROUS JUSTIFICATION OF THE FOKKER-PLANCK EQUATIONS OF NEURAL NETWORKS BASED ON AN ITERATION PERSPECTIVE. SIAM Journal on Mathematical Analysis. 2022 Jan 1;54(1):1270–1312.

Published In

SIAM Journal on Mathematical Analysis

DOI

EISSN

1095-7154

ISSN

0036-1410

Publication Date

January 1, 2022

Volume

54

Issue

1

Start / End Page

1270 / 1312

Related Subject Headings

  • Applied Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics