MCMC-Net: Accelerating Markov Chain Monte Carlo with Neural Networks for Inverse Problems
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912585162424320 |
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| author | Majee, Sudeb Abhishek, Anuj Strauss, Thilo Khan, Taufiquar |
| author_facet | Majee, Sudeb Abhishek, Anuj Strauss, Thilo Khan, Taufiquar |
| contents | In many computational problems, using the Markov Chain Monte Carlo (MCMC) can be prohibitively time-consuming. We propose MCMC-Net, a simple yet efficient way to accelerate MCMC via neural networks. The key idea of our approach is to substitute the true likelihood function of the MCMC method with a neural operator based surrogate. We extensively evaluate the accuracy and speedup of our method on three different PDE-based inverse problems where likelihood computations are computationally expensive, namely electrical impedance tomography, diffuse optical tomography, and quantitative photoacoustic tomography.
MCMC-Net performs similar to the classical likelihood counterpart but with a significant speedup. We conjecture that the method can be applied to any problem with a sufficiently expensive likelihood function. We also analyze MCMC-Net in a theoretical setting for the different use cases. We prove a universal approximation theorem-type result to show that the proposed network can approximate the mapping resulting from forward model evaluations to a desired accuracy. Furthermore, we establish convergence of the surrogate posterior to the true posterior under Hellinger distance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16883 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | MCMC-Net: Accelerating Markov Chain Monte Carlo with Neural Networks for Inverse Problems Majee, Sudeb Abhishek, Anuj Strauss, Thilo Khan, Taufiquar Numerical Analysis 62G20, 35J15, 62F15 In many computational problems, using the Markov Chain Monte Carlo (MCMC) can be prohibitively time-consuming. We propose MCMC-Net, a simple yet efficient way to accelerate MCMC via neural networks. The key idea of our approach is to substitute the true likelihood function of the MCMC method with a neural operator based surrogate. We extensively evaluate the accuracy and speedup of our method on three different PDE-based inverse problems where likelihood computations are computationally expensive, namely electrical impedance tomography, diffuse optical tomography, and quantitative photoacoustic tomography. MCMC-Net performs similar to the classical likelihood counterpart but with a significant speedup. We conjecture that the method can be applied to any problem with a sufficiently expensive likelihood function. We also analyze MCMC-Net in a theoretical setting for the different use cases. We prove a universal approximation theorem-type result to show that the proposed network can approximate the mapping resulting from forward model evaluations to a desired accuracy. Furthermore, we establish convergence of the surrogate posterior to the true posterior under Hellinger distance. |
| title | MCMC-Net: Accelerating Markov Chain Monte Carlo with Neural Networks for Inverse Problems |
| topic | Numerical Analysis 62G20, 35J15, 62F15 |
| url | https://arxiv.org/abs/2412.16883 |