Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918202585382912 |
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| author | Karris, Nicholas Nikitopoulos, Evangelos A. Kevrekidis, Ioannis G. Lee, Seungjoon Cloninger, Alexander |
| author_facet | Karris, Nicholas Nikitopoulos, Evangelos A. Kevrekidis, Ioannis G. Lee, Seungjoon Cloninger, Alexander |
| contents | We develop an Euler-type method to predict the evolution of a time-dependent probability measure without explicitly learning an operator that governs its evolution. We use linearized optimal transport theory to prove that the measure-valued analog of Euler's method is first-order accurate when the measure evolves ``smoothly.'' In applications of interest, however, the measure is an empirical distribution of a system of stochastic particles whose behavior is only accessible through an agent-based micro-scale simulation. In such cases, this empirical measure does not evolve smoothly because the individual particles move chaotically on short time scales. However, we can still perform our Euler-type method, and when the particles' collective distribution approximates a measure that \emph{does} evolve smoothly, we observe that the algorithm still accurately predicts this collective behavior over relatively large Euler steps, thus reducing the number of micro-scale steps required to step forward in time. In this way, our algorithm provides a ``macro-scale timestepper'' that requires less micro-scale data to still maintain accuracy, which we demonstrate with three illustrative examples: a biological agent-based model, a model of a PDE, and a model of Langevin dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01857 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems Karris, Nicholas Nikitopoulos, Evangelos A. Kevrekidis, Ioannis G. Lee, Seungjoon Cloninger, Alexander Numerical Analysis Machine Learning Optimization and Control We develop an Euler-type method to predict the evolution of a time-dependent probability measure without explicitly learning an operator that governs its evolution. We use linearized optimal transport theory to prove that the measure-valued analog of Euler's method is first-order accurate when the measure evolves ``smoothly.'' In applications of interest, however, the measure is an empirical distribution of a system of stochastic particles whose behavior is only accessible through an agent-based micro-scale simulation. In such cases, this empirical measure does not evolve smoothly because the individual particles move chaotically on short time scales. However, we can still perform our Euler-type method, and when the particles' collective distribution approximates a measure that \emph{does} evolve smoothly, we observe that the algorithm still accurately predicts this collective behavior over relatively large Euler steps, thus reducing the number of micro-scale steps required to step forward in time. In this way, our algorithm provides a ``macro-scale timestepper'' that requires less micro-scale data to still maintain accuracy, which we demonstrate with three illustrative examples: a biological agent-based model, a model of a PDE, and a model of Langevin dynamics. |
| title | Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems |
| topic | Numerical Analysis Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2408.01857 |