X-CAL: Explaining latent causality in physical space for fluid mechanics

TL;DR

X-CAL introduces a principled, explainable pipeline that unites nonlinear latent compression ($β$-VAE), information-theoretic causality (SURD), and gradient-SHAP explainability to reveal causal interactions among latent flow features and map them to physical-space structures. Validated on DNS data for flow around a wall-mounted square cylinder at $Re_h=2000$ and demonstrated in controlled 2D torus and Lorenz systems, the framework identifies how latent variables causally influence each other and which wake structures drive those dynamics. The results show latent variables capture coherent wake regions, with a co-founder latent ($\mathcal{L}_3$) connecting subdomains and orchestrating vortex-shedding dynamics, offering a pathway to causality-guided control and discovery in high-dimensional turbulence. Overall, X-CAL translates latent-space causality into interpretable physical phenomena, enabling robust reduced-order modeling and targeted interventions in complex fluid flows.

Abstract

We present X-CAL, a pipeline that combines a $β$-variational autoencoder ($β$-VAE) with the synergistic-unique-redundant decomposition (SURD)~\cite{surd} approach for causality analysis to interpret low-dimensional latent representations of turbulent fluid flows. Combining $β$-VAE compression with SURD and SHAP (SHapley Additive exPlanations) yields interpretable latent representations and structure-level attributions in physical space, offering a general methodology for causal analysis of high-dimensional flows. Using direct numerical simulation (DNS) data of the flow around a wall-mounted square cylinder at $Re_h=2000$, we (i) learn a compact latent space with near-orthogonal variables, (ii) quantify directed information flows among these variables via the SURD approach, and (iii) map latent-space causality back to physical space through gradient-SHAP fields . By means of percolation analysis of the SHAP fields, we extract the coherent, time-resolved structures that most influence each latent variable. The analysis connects coherent structures with latent variables which are in turn associated with wake-boundary-layer interactions. This method enables translating the insight obtained through causal analysis in the latent space into interpretable phenomena in physical space.

Figures (30)

• Figure 1: Causal artificial-intelligence (AI) framework visualization. (Left) Instantaneous flow field of the case under study. (Right) Schematic representation of the method, where 1) we encode the flow field into a latent space with a $\beta$-VAE; 2) we analyze causality among the latent variables, identifying key causal mechanisms; 3) we perform SHAP analysis to identify the most important flow structures associated with each of the latent variables. This enables formulating the causal relations identified in the latent space in terms of coherent structures in physical space.
• Figure 2: (Left) First lag phase plot between temporal coefficients of the system and (Right) the SURD decomposition for the temporal sequences $f_1,f_2$.
• Figure 3: (Left) SURD decomposition analysis for all cases presented in Table \ref{['torus_tab']} and the reference temporal signal in Fig \ref{['torus_info']}, the different terms in the decomposition are plotted with their respective magnitude as a function of the $\beta_n$, where the numbers on the $x$-axis refer to the study cases and (Right) First lag phase plot between latent temporal coefficients of Case IV.
• Figure 4: Spatial basis functions used to lift the Lorenz trajectories into 2D fields: (a) $\phi_1$, (b) $\phi_2$, (c) $\phi_3$.
• Figure 5: First we inspect the Mutual information between the original temporal coefficients of the Lorenz System $x_t,y_t,z_t$ and the latent coefficients for Case I, $s_1,s_2,s_3$ (left), secondly we inspect the mutual information in between Lorenz coefficients (middle) and finally we compute the mutual information between the latent elements of Case I (right).