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

This paper introduces X-CAL, a pipeline combining $\beta$-VAE, SURD, and SHAP to interpret low-dimensional latent representations of turbulent flows by quantifying causal interactions and mapping them back to coherent physical structures in wall-mounted cylinder flows.

The Gist

Imagine you are trying to understand a chaotic, swirling storm of water flowing around a square pillar stuck in a river. To the naked eye, it looks like a messy, unpredictable mess of eddies and currents. Scientists have long known that this mess is actually made of specific, repeating shapes (like spinning vortices), but figuring out how one shape causes another to appear, and why they interact the way they do, is like trying to understand a complex machine by just watching the smoke come out of the chimney.

This paper introduces a new tool called X-CAL to solve this puzzle. Think of X-CAL as a "causal detective" that uses artificial intelligence to translate the messy, high-speed physics of the water into a simple, understandable story.

Here is how X-CAL works, broken down into three simple steps using everyday analogies:

1. The Compression: Turning a Symphony into a Playlist

The flow of water around the pillar is incredibly complex, with millions of data points moving every second. It's like trying to listen to a 100-piece orchestra playing a symphony all at once; it's too much information to process.

X-CAL first uses a special AI brain (called a $\beta$-VAE) to act as a "music producer." This producer listens to the entire chaotic symphony and compresses it down into just three simple notes (called "latent variables").

• The Magic Trick: Unlike older methods that just pick the loudest notes, this AI is trained to make sure these three notes are distinct and don't overlap. It forces them to be "near-orthogonal," which is a fancy way of saying it ensures each note represents a completely different part of the story, so they don't confuse each other.

2. The Detective Work: Figuring Out Who Influences Whom

Now that the complex flow is reduced to three simple notes, the researchers need to know: Does Note A cause Note B? Or does Note B cause Note A?

To answer this, they use a mathematical method called SURD. Imagine you are watching a game of telephone.

• Unique Causality: This is when one person (Note A) whispers a secret that only they know, and it directly changes what the next person (Note B) says.
• Redundant Causality: This is when two people (Note A and Note C) both whisper the same secret to Note B.
• Synergistic Causality: This is when Note A and Note C whisper different things, but only when you hear them together does Note B understand the full message.

X-CAL uses this logic to map out a "family tree" of cause and effect between the three notes. It tells the researchers exactly which "note" is driving the others and when.

3. The Translation: Mapping the Notes Back to the River

The final step is the most important. The researchers have a map of how the three "notes" influence each other, but they need to know what those notes look like in the actual river.

They use a tool called SHAP (which acts like a "highlighter pen").

• The AI asks: "Which specific drops of water in the river were most responsible for creating 'Note A'?"
• The highlighter marks those specific areas. By looking at these highlighted areas, the researchers can see that "Note A" isn't just a number; it's actually a swirling vortex forming near the bottom of the pillar. "Note B" might be a shear layer (a thin sheet of fast-moving water) near the top.

What Did They Discover?

By applying X-CAL to a computer simulation of water flowing around a square pillar, the researchers found a clear, causal chain of events:

1. The Trigger: A vortex forms at the very top tip of the pillar (the "tip vortex").
2. The Chain Reaction: This top vortex doesn't just sit there; it travels downstream and causes a specific change in the water flow near the bottom of the pillar.
3. The Cycle: This interaction causes the bottom vortex to lift up and mix with the top flow, eventually leading to a new vortex shedding (falling off) from the top again.

The Big Picture:
The paper shows that X-CAL can take a chaotic, high-dimensional mess of fluid physics, compress it into a few understandable "characters," figure out the script of how those characters interact, and then translate that script back into a visual map of the actual water flow.

Instead of just saying "the flow is turbulent," X-CAL allows scientists to say: "The top vortex causes the bottom vortex to lift up, which then triggers the next cycle of shedding." This turns a blurry picture of chaos into a clear, causal story that engineers can use to understand and eventually control these flows.

Technical Summary

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

Problem Statement
Turbulent fluid flows are governed by nonlinear Navier–Stokes equations, resulting in high-dimensional, complex systems where the dynamic interactions and causal dependencies of coherent structures (e.g., vortices, shear layers) remain difficult to quantify. While traditional reduced-order modeling (ROM) techniques like Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) effectively identify dominant energetic structures, their linear formulations fail to capture explicit nonlinear causal links. Consequently, these methods cannot quantify directed information exchange or explain why specific structures influence subsequent flow evolution. There is a critical need for a methodology that combines dimensionality reduction with explicit causal analysis and physical interpretability to enable robust predictive capability and effective flow control.

Methodology: The X-CAL Framework
The authors present X-CAL (Explainable causal analysis for latent representations), a pipeline integrating three core components to bridge latent space causality with physical space phenomena:

1. Nonlinear Compression ($\beta$-VAE): The framework employs a $\beta$-Variational Autoencoder ($\beta$-VAE) to compress high-dimensional flow fields into a low-dimensional, disentangled latent manifold. The $\beta$-regularization term encourages latent variables to be near-orthogonal and minimizes redundancy, creating an interpretable latent space where causal relations can be analyzed more clearly than in the raw physical domain.
2. Explicit Causality Analysis (SURD): Directed information flow among the disentangled latent variables is quantified using the Synergistic–Unique–Redundant (SURD) decomposition. This approach, rooted in information theory, decomposes mutual information into:
   • Unique causality: Information provided exclusively by a single source variable.
   • Redundant causality: Information shared identically across a group of variables.
   • Synergistic causality: Information accessible only when variables are considered jointly.
   • Causality leak: Information from unobserved variables.
     This allows for the identification of specific causal mechanisms and state-dependent interactions.
3. Physical Attribution (Gradient-SHAP and Percolation): To map latent causality back to physical space, the framework uses Gradient-SHAP (SHapley Additive exPlanations). This generates spatially explicit fields quantifying how each grid point in the input flow contributes to the encoding of specific latent variables. By applying percolation analysis to these SHAP fields, the method extracts coherent, time-resolved structures that drive the dynamics, yielding structure-level attributions.

Validation and Results
The framework was first validated on synthetic chaotic systems (a 2D Torus and the Lorenz system) to ensure the $\beta$-VAE successfully preserves or learns key causal signatures. In the Lorenz system, the framework demonstrated the ability to disentangle causal structures and identify unique and synergistic contributions that were not apparent in the original variables.

The primary application was applied to Direct Numerical Simulation (DNS) data of flow around a wall-mounted square cylinder at $Re_h = 2000$. Key findings include:

• Latent Space Compression: A 3-dimensional latent space was learned, achieving high reconstruction accuracy (93.5–94.4% for streamwise and vertical fluctuations) while maintaining near-orthogonal latent variables.
• Causal Mechanisms: SURD analysis revealed that latent variable $L_3$ acts as a dominant source of unique information for $L_1$ and $L_2$. The analysis identified specific time lags corresponding to physical shedding frequencies.
• Physical Interpretation:
  • $L_1$ is associated with the near-low wake and base vortex dynamics, driven by the tip shear layer convecting downstream.
  • $L_2$ captures the transition from a quasi-parallel upper shear layer to localized shedding near the obstacle tip (initiation of a new vortex street).
  • $L_3$ represents the complete tip-vortex shedding cycle and connects different subdomains of the wake, acting as a "co-founder" variable that synchronizes tip-layer instabilities and spanwise shedding.
• Causal Chains: The study mapped causal events, such as the tip vortex forming at the obstacle tip (cause) and subsequently influencing the low-wake recirculation bubble (effect), demonstrating how latent causality translates to physical coherent structures.

Significance and Claims
The paper claims that X-CAL provides a general methodology for causal analysis in high-dimensional flows by translating insights from latent space into interpretable physical phenomena. The significance of the work lies in:

• Objective Importance: Unlike traditional control strategies based on ad-hoc assumptions, X-CAL defines the importance of flow structures objectively via SHAP values, identifying the specific regions and mechanisms that govern flow dynamics.
• Causal-AI Integration: It demonstrates that integrating causality and explainability into machine learning architectures increases trust and utility for scientific discovery and flow control.
• Mechanism Identification: The framework successfully connects latent causal dependencies to specific coherent structures (tip vortices, base vortices, spanwise shedding), offering a pathway to design control strategies targeting specific mechanisms rather than global statistics.
• Future Applicability: The authors suggest the framework can enhance the understanding of closure models in reduced-order modeling and could be extended to 3D datasets to identify the full set of coherent motions (including horseshoe vortices) in turbulent wakes.