提出新准则PASTIS，从噪声数据中自动选出最简正确随机模型。
Model selection for stochastic dynamics: a parsimonious and principled approach
- 基于极值理论设计惩罚项，融合候选模型规模与显著性阈值。
- 在洛伦兹、格雷-斯科特等系统上，准确率和预测能力均优于经典方法。
- 支持高采样间隔和强噪声数据，适合真实实验场景建模。
本论文聚焦于从含噪离散时间序列中发现随机微分方程（SDE）与随机偏微分方程（SPDE）。核心挑战是从海量候选模型中选取最简正确的模型，而传统信息准则（AIC、BIC）常受限。本文提出PASTIS（Parsimonious Stochastic Inference），一种基于极值理论的新信息准则。其惩罚项 $n_/mathcal{B} \ln(n_0/p)$ 显式包含初始候选参数库大小 $n_0$、所选模型参数数 $n_/mathcal{B}$ 与显著性阈值 $p$，其中 $p$ 表示在多模型比较中误选冗余参数的概率。在洛伦兹、奥恩斯坦-乌伦贝克、洛特卡-沃尔泰拉（SDE）及格雷-斯科特（SPDE）系统上的基准测试表明，PASTIS 在精确模型识别与预测能力上均优于AIC、BIC、交叉验证（CV）和SINDy。针对实际数据中大采样间隔（Δt）或高测量噪声（σ）带来的建模困难，本文进一步构建了鲁棒变体PASTIS-Δt与PASTIS-σ，显著拓展了方法在不完美实验数据中的适用性。PASTIS提供了一套统计严谨、验证充分且实用的随机动力学建模框架。
原文摘要 · Abstract (English)
This thesis focuses on the discovery of stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) from noisy and discrete time series. A major challenge is selecting the simplest possible correct model from vast libraries of candidate models, where standard information criteria (AIC, BIC) are often limited. We introduce PASTIS (Parsimonious Stochastic Inference), a new information criterion derived from extreme value theory. Its penalty term, $n_\mathcal{B} \ln(n_0/p)$, explicitly incorporates the size of the initial library of candidate parameters ($n_0$), the number of parameters in the considered model ($n_\mathcal{B}$), and a significance threshold ($p$). This significance threshold represents the probability of selecting a model containing more parameters than necessary when comparing many models. Benchmarks on various systems (Lorenz, Ornstein-Uhlenbeck, Lotka-Volterra for SDEs; Gray-Scott for SPDEs) demonstrate that PASTIS outperforms AIC, BIC, cross-validation (CV), and SINDy (a competing method) in terms of exact model identification and predictive capability. Furthermore, real-world data can be subject to large sampling intervals ($Δt$) or measurement noise ($σ$), which can impair model learning and selection capabilities. To address this, we have developed robust variants of PASTIS, PASTIS-$Δt$ and PASTIS-$σ$, thus extending the applicability of the approach to imperfect experimental data. PASTIS thus provides a statistically grounded, validated, and practical methodological framework for discovering simple models for processes with stochastic dynamics.
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