LangevinFlow: Stochastic Modeling Approach

• LangevinFlow is a data-driven stochastic framework based on the nonlinear Langevin equation that extracts evolution equations from empirical measurements.
• It applies conditional statistics and the Markov property to simplify complex, multiscale phenomena, notably in turbulent flows.
• Its parameter-free approach is extendable to diverse fields like finance, medicine, and geophysics for modeling time and scale dynamics.

LangevinFlow refers to a class of methodologies and analytical frameworks grounded in the nonlinear Langevin equation, designed to extract evolution equations for stochastic variables directly from empirical measurements. Its principal innovation is the construction of parameter-free, data-driven stochastic models—most notably applicable to complex systems exhibiting multiscale phenomena, such as turbulence—by leveraging conditional statistics and the Markov property. The approach extends beyond classical temporal evolution, enabling the modeling of both time-based and scale-based processes, provided the underlying data exhibit ergodicity.

1. Mathematical Foundations of the Langevin Approach

The underlying mathematical object of LangevinFlow is the nonlinear Langevin equation: $$\frac{dX_i}{dt} = F_i(X_1,\ldots,X_N, t) + G_i(X_1,\ldots,X_N, \{\xi_j\}),$$ where $F_i$ characterizes deterministic dynamics and $G_i$ accounts for stochastic forcing via noise terms $\xi_j$.

For scale-dependent processes (where the scale variable replaces time), increments are defined as: $\Delta X(x, r) = X(x+r) - X(x),$ with scale $r$ (or, for additive evolution, $s = \log r$). The probability evolution of increments is described through the scale-propagator: $$p\big(\Delta X(s+\Delta s)\mid \Delta X(s), \Delta X(s-\Delta s), \dots\big).$$ Assuming the Markov property (i.e., future increments depend only on present values), the system's multipoint statistics reduce to two-point statistics, allowing the application of the Kramers–Moyal (KM) expansion: $$\frac{\partial}{\partial s} p(\Delta X,s) = \sum_{k\ge1} \left[-\frac{\partial}{\partial \Delta X}\right]^k \left[D^{(k)}(\Delta X,s) p(\Delta X,s)\right].$$ The KM coefficients are identified as: $$D^{(k)}(\Delta X,s) = \lim_{\Delta s \to 0} \frac{1}{k! \, \Delta s} M^{(k)}(\Delta X, \Delta s),$$ with the conditional moments

$$M^{(k)}(\Delta X, s, \Delta s) = \int (Y-\Delta X)^k\, p(Y \mid \Delta X) dY.$$

If the fourth-order coefficient $D^{(4)}$ vanishes (per Pawula's theorem), the expansion truncates to the Fokker–Planck equation: $$\frac{\partial}{\partial s} p(\Delta X,s) = -\frac{\partial}{\partial \Delta X} \left[ D^{(1)}(\Delta X,s) p(\Delta X, s) \right] + \frac{\partial^2}{\partial \Delta X^2} \left[ D^{(2)}(\Delta X, s) p(\Delta X, s) \right].$$ This equation is equivalent to a Langevin equation in scale variable $s$: $$\frac{d\Delta X}{ds} = D^{(1)}(\Delta X,s) + \sqrt{D^{(2)}(\Delta X,s)}\,\eta(s),$$ where $\eta(s)$ is a delta-correlated noise process.

2. Numerical Implementation Workflow

The practical realization of LangevinFlow involves multiple clearly segmented steps:

• Data Preparation and Precondition Verification: The methodology requires that the observed process is stationary and the increments obey the Markov property. Stationarity is typically verified by evaluating the invariance of central moments across data subsets, while the Markov character is tested both qualitatively (via conditional pdf contour plots) and quantitatively (using the Wilcoxon test or Kullback–Leibler divergence).
• Estimation of Kramers–Moyal Coefficients: Conditional moments

$M^{(k)}(\Delta X, s, \Delta s)$

are computed across bins of increment values and for a range of small $\Delta s$. A linear relationship between moments and $\Delta s$ enables the extraction of the coefficient as the slope; direct evaluation at the smallest $\Delta s$ is also viable.

• Uncertainty and Model Optimization: Errors on the coefficients are estimated based on:

$$\delta M^{(k)}(\Delta X,\Delta s) = M^{(2k)}(\Delta X,\Delta s) - [M^{(k)}(\Delta X, \Delta s)]^2.$$

Optimization is performed by refining $D^{(1)}$ and $D^{(2)}$ such that reconstructed conditional pdfs match empirical statistics.

This approach is parameter-free, extracting the evolution equations directly from measurements without the need for fitting model parameters in advance.

3. Application to Turbulent Velocity Fields

The LangevinFlow methodology is concretely applied to turbulent velocity field data from both laboratory experiments and computational simulations. For experiments, velocity fields are typically recorded in wind-tunnel setups with hot-wire anemometry at high frequencies and over long durations. The turbulent cascade is addressed as a scale process: energy introduced at large scales (large eddies) cascades down to dissipative small scales.

In simulations, Delayed Detached Eddy Simulation (DDES) is used, with OpenFOAM providing computationally generated velocity fields under conditions closely mirroring the experimental apparatus. The grid geometry is matched precisely.

The LangevinFlow approach extracts drift ($D^{(1)}$) and diffusion ($D^{(2)}$) terms as functions of scale. Key results include:

• Strong correspondence between experimental and simulated dominant coefficients (e.g., linear scaling of $d_{11}$ and consistency in $d_{20}$ offsets), which quantitatively represent the growth of incremental velocity (or eddy strength) with scale.
• Discrepancies in higher-order terms such as $d_{22}$ are observed and noted as requiring further examination.

This analysis validates the capacity of the LangevinFlow method to reconstruct the turbulent energy cascade and its multiscale dynamics from observables.

4. Physical Interpretation: Linking Time and Scale

A central insight provided by the LangevinFlow framework is the mapping of stochastic time evolution to scale evolution (i.e., in the logarithmic scale variable $s = \log r$). This correspondence is physically motivated:

• Galton Box Analogy: Analogous to Brownian motion, the random walk process in scale is conceptualized such that, across discrete “rows” (scale increments), the distribution of increments approaches a Gaussian with variance proportional to the scale increment.
• Linking Time-Series Reconstruction: Because statistical properties in scale conform to Markovian structure under ergodicity, the full time evolution of a process can be reconstructed or forecasted from the scale-propagator, thus bridging scale-based and time-based process analysis.

This perspective elucidates the physical foundations of the turbulent cascade and supports the reduction of complex dynamics to a small set of informative stochastic parameters.

5. Extensions and Broader Applications

LangevinFlow has demonstrated utility far beyond turbulence, being adaptable to:

• Time Series Reconstruction: Leveraging the connection between scale propagation and time propagation, the method can reconstruct or predict temporal data, including scenarios where the original measurement is noisy or incomplete.
• Finance: Modeling financial time series using reduced-order stochastic processes identified by LangevinFlow.
• Medicine: Analyzing heart rhythms and brain signal dynamics as driven by stochastic evolution equations.
• Geophysics: Application to the analysis of seismic and earth-system signals, characterized by multi-scale variability.
• Renewable Energies: Interpreting wind energy converter data within the Langevin-based stochastic modeling framework.

The main strengths of LangevinFlow include its parameter-free extraction of evolution equations, capability for physical insight (especially in synthesizing the energy cascade), and its reduction of high-dimensionality complex phenomena to tractable low-dimensional representations.

However, effective implementation is contingent upon the data meeting strict criteria: stationarity and the Markov property. Where these fail (due to, for example, nonstationarity or pronounced measurement noise), the approach may require adaptation or extension.

6. Significance and Outlook

LangevinFlow offers a powerful tool for modeling and analyzing complex phenomena that are inherently stochastic and multi-scale in nature. By focusing on conditional statistics and their evolution across either time or scale, the method facilitates both the understanding and prediction of crucial features such as turbulent cascades in fluids, patterns in financial time series, or physiological rhythms. Its foundations in the stochastic nonlinear Langevin equation and the rigorous extraction of equations from empirical data make it broadly applicable while preserving a strong connection to the physical underpinnings of the studied system (Reinke et al., 2015).