Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916330693722112 |
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| author | Galioto, Nicholas Sharma, Harsh Kramer, Boris Gorodetsky, Alex Arkady |
| author_facet | Galioto, Nicholas Sharma, Harsh Kramer, Boris Gorodetsky, Alex Arkady |
| contents | This paper presents a structure-preserving Bayesian approach for learning nonseparable Hamiltonian systems using stochastic dynamic models allowing for statistically-dependent, vector-valued additive and multiplicative measurement noise. The approach is comprised of three main facets. First, we derive a Gaussian filter for a statistically-dependent, vector-valued, additive and multiplicative noise model that is needed to evaluate the likelihood within the Bayesian posterior. Second, we develop a novel algorithm for cost-effective application of Bayesian system identification to high-dimensional systems. Third, we demonstrate how structure-preserving methods can be incorporated into the proposed framework, using nonseparable Hamiltonians as an illustrative system class. We assess the method's performance based on the forecasting accuracy of a model estimated from single-trajectory data. We compare the Bayesian method to a state-of-the-art machine learning method on a canonical nonseparable Hamiltonian model and a chaotic double pendulum model with small, noisy training datasets. The results show that using the Bayesian posterior as a training objective can yield upwards of 724 times improvement in Hamiltonian mean squared error using training data with up to 10% multiplicative noise compared to a standard training objective. Lastly, we demonstrate the utility of the novel algorithm for parameter estimation of a 64-dimensional model of the spatially-discretized nonlinear Schrödinger equation with data corrupted by up to 20% multiplicative noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling Galioto, Nicholas Sharma, Harsh Kramer, Boris Gorodetsky, Alex Arkady Machine Learning Dynamical Systems Data Analysis, Statistics and Probability Computation This paper presents a structure-preserving Bayesian approach for learning nonseparable Hamiltonian systems using stochastic dynamic models allowing for statistically-dependent, vector-valued additive and multiplicative measurement noise. The approach is comprised of three main facets. First, we derive a Gaussian filter for a statistically-dependent, vector-valued, additive and multiplicative noise model that is needed to evaluate the likelihood within the Bayesian posterior. Second, we develop a novel algorithm for cost-effective application of Bayesian system identification to high-dimensional systems. Third, we demonstrate how structure-preserving methods can be incorporated into the proposed framework, using nonseparable Hamiltonians as an illustrative system class. We assess the method's performance based on the forecasting accuracy of a model estimated from single-trajectory data. We compare the Bayesian method to a state-of-the-art machine learning method on a canonical nonseparable Hamiltonian model and a chaotic double pendulum model with small, noisy training datasets. The results show that using the Bayesian posterior as a training objective can yield upwards of 724 times improvement in Hamiltonian mean squared error using training data with up to 10% multiplicative noise compared to a standard training objective. Lastly, we demonstrate the utility of the novel algorithm for parameter estimation of a 64-dimensional model of the spatially-discretized nonlinear Schrödinger equation with data corrupted by up to 20% multiplicative noise. |
| title | Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling |
| topic | Machine Learning Dynamical Systems Data Analysis, Statistics and Probability Computation |
| url | https://arxiv.org/abs/2401.12476 |