🤖 machine learning

Power-scaled Bayesian Inference with Score-based Generative Models

This paper proposes a score-based generative algorithm that enables flexible, retraining-free control over prior-likelihood influence in Bayesian inference for seismic velocity modeling, demonstrating that power-scaling the likelihood improves data fidelity while reducing prior power enhances structural diversity.

Original authors: Huseyin Tuna Erdinc, Yunlin Zeng, Abhinav Prakash Gahlot, Felix J. Herrmann

Published 2026-04-03

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CC BY 4.0

Original authors: Huseyin Tuna Erdinc, Yunlin Zeng, Abhinav Prakash Gahlot, Felix J. Herrmann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to draw a detailed map of the underground world (where oil and gas are hidden), but you can't see it directly. You only have blurry, noisy photos taken from the surface (seismic images). To fill in the gaps, you need to guess what the underground looks like.

This paper introduces a clever new way to make those guesses using Artificial Intelligence (AI) and a concept called "Power-Scaling."

Here is the breakdown using simple analogies:

1. The Problem: The "Too Strict" vs. "Too Wild" Dilemma

In the past, scientists trying to map the underground had two bad options:

Option A (Too Strict): They relied heavily on a "rulebook" (called a Prior). This rulebook said, "The ground usually looks like this." If the data didn't match the rulebook, they ignored the data. The result? A map that looked perfect but might be wrong because it ignored the actual clues.
Option B (Too Wild): They ignored the rulebook and trusted only the noisy photos (the Likelihood). The result? A map that matched the photos perfectly but looked like a chaotic mess with no logical geological layers.

They needed a way to balance the two: "Listen to the data, but keep the map looking geologically sensible."

2. The Solution: The "Volume Knob" (Power-Scaling)

The authors built a smart AI that acts like a mixing board with volume knobs.

Instead of training a new AI every time they wanted to change the balance, they trained one single AI that can generate both the "Rulebook" (what the ground usually looks like) and the "Photo Match" (what the ground actually looks like based on data).

Then, they introduced Power-Scaling. Think of this as turning the volume up or down on the two inputs:

The Prior Knob (α): Controls how much the AI sticks to the "Rulebook" (geological structure).
The Likelihood Knob (λ): Controls how much the AI listens to the "Photos" (the actual seismic data).

3. How It Works in Practice

The researchers tested this by turning the knobs to see what happened:

Turning down the Prior Knob (Low α):
- Analogy: Taking off the training wheels.
- Result: The AI generates many different, diverse maps. Some look weird, but it explores many possibilities. It's great for seeing "what if" scenarios.
Turning up the Prior Knob (High α):
- Analogy: Putting the training wheels back on tight.
- Result: The maps become very neat, with sharp, clear layers. They look very "geological" but might ignore some specific details in the data.
Turning up the Likelihood Knob (High λ):
- Analogy: Cranking up the volume on the radio so you hear the song perfectly, even if the static is loud.
- Result: The map matches the seismic photos very closely.
- The Surprise: The researchers found that turning the Likelihood knob up higher than normal (about double the usual setting) actually made the map more accurate than standard methods. It was like finding the "sweet spot" where the AI ignored the noise in the photos but kept the important details.

4. The "Compass" Visualization

The authors created a "Power-Scaling Compass." Imagine a grid:

Left side: Maps that look very structured (High Prior).
Right side: Maps that look very chaotic (Low Prior).
Bottom: Maps that ignore the data (Low Likelihood).
Top: Maps that follow the data strictly (High Likelihood).

By moving around this compass, a scientist can instantly see how much the final map is influenced by the "Rulebook" vs. the "Photos."

Why Is This a Big Deal?

No Retraining: Usually, if you want to change how an AI balances data and rules, you have to retrain it from scratch (which takes days and costs a lot of money). This method lets you change the balance instantly just by turning a virtual knob.
Better Maps: They found that by slightly "over-trusting" the data (turning the Likelihood knob up), they could get cleaner, more accurate underground maps than before.
Safety Check: It allows scientists to do a "sensitivity analysis." They can ask, "What if the data is wrong? What if the rulebook is wrong?" and see how the map changes, helping them understand the uncertainty of their results.

Summary

Think of this paper as inventing a universal remote control for underground mapping. Instead of building a new camera for every different lighting condition, you just have one camera and a remote that lets you adjust the brightness and contrast instantly. This helps geologists create better maps of the Earth's interior without the expensive wait of retraining their AI models.

1. Problem Statement

The paper addresses critical challenges in subsurface velocity model generation, a key task in geophysical exploration (e.g., hydrocarbon exploration).

Limitations of Traditional Methods: Full-Waveform Inversion (FWI), while high-resolution, is computationally expensive, prone to local minima, and highly sensitive to initial models due to the non-linear and ill-posed nature of the inverse problem.
Limitations of Current Generative Approaches: Recent methods using generative models often rely on strong, structured priors that can dominate the inference, limiting the model's responsiveness to observed data. Conversely, methods using non-informative priors lack sufficient regularization, leading to instability in noisy or underdetermined regimes.
The Core Gap: There is a need for a flexible framework that allows explicit control over the trade-off between prior regularization (structural consistency) and likelihood data fidelity without requiring retraining for different configurations.

2. Methodology

The authors propose a novel framework combining Bayesian inference with Score-Based Generative Models (SBGMs) using Power-Scaling.

A. Theoretical Framework

Bayesian Formulation: The goal is to sample from the posterior $p(x|y)$ , where $x$ is the velocity model and $y$ is the observed data (specifically, Reverse-Time Migration (RTM) images used as summary statistics).
Power-Scaled Posterior: The authors introduce a power-scaled posterior distribution:
$p_{\lambda,\alpha}(x|y) \propto p(y|x)^\lambda p(x)^\alpha$
Where $\lambda$ scales the likelihood and $\alpha$ scales the prior. This allows for amplifying or attenuating the influence of data and prior assumptions independently.
Score-Based Estimation: Instead of training separate networks for the prior and posterior, the method leverages a single conditional score-based network trained with Classifier-Free Guidance (CFG).
- During training, the conditioning input (RTM image) is randomly masked (dropped) with probability $p=0.2$ .
- This enables the network to learn both the unconditional score ( $\nabla_x \log p(x)$ ) and the conditional score ( $\nabla_x \log p(x|y)$ ) simultaneously.
Inference Strategy:
- Using the identity $\nabla_x \log p(y|x) = \nabla_x \log p(x|y) - \nabla_x \log p(x)$ , the power-scaled score is constructed as a linear combination:
  $\nabla_x \log p_{\lambda,\alpha}(x|y) = \lambda \nabla_x \log p(x|y) + (\alpha - \lambda) \nabla_x \log p(x)$
- Samples are generated by solving a Stochastic Differential Equation (SDE) or Ordinary Differential Equation (ODE) using this combined score.
- Langevin Correction: To ensure accuracy, a predictor-corrector step (Langevin dynamics) is incorporated into the sampling loop.

B. Data and Setup

Dataset: A synthetic 3D Earth model (Compass model, North Sea region) sliced into 2D velocity models ( $256 \times 512$ grid).
Conditioning Data: RTM images generated from shot records (16 sources, 256 receivers, 20Hz Ricker wavelet) with added noise.
Architecture: A U-Net trained to denoise velocity samples conditioned on RTM images.

3. Key Contributions

Flexible Power-Scaling Framework: A method to sample from power-scaled priors and likelihoods using a single trained network, eliminating the need for retraining when adjusting the prior-likelihood balance.
Efficient Surrogate Training: Utilization of Classifier-Free Guidance to estimate both prior and posterior scores within a single network, avoiding the computational cost of training separate models.
Avoidance of Explicit Likelihood Evaluation: The method bypasses the need for expensive, repeated wave-equation solves (PDEs) during inference, as the likelihood influence is encoded in the learned score function.
Sensitivity Analysis Tool: The framework enables "Power-Scaling Compass" experiments to visualize and quantify the trade-offs between data fidelity and geological structural regularity.

4. Experimental Results

The authors conducted experiments on synthetic data to evaluate the effects of varying $\lambda$ (likelihood power) and $\alpha$ (prior power).

Prior Scaling ( $\alpha$ ):
- Low $\alpha$ ( $<1$ ): Results in high variability and diffuse geological structures, effectively relaxing constraints.
- High $\alpha$ ( $>1$ ): Produces sharper, more regularized layers, reinforcing the geological patterns learned from the training data.
Likelihood Scaling ( $\lambda$ ):
- Standard Bayesian ( $\lambda=1$ ): Provides moderate data fidelity.
- Optimal Scaling ( $\lambda \approx 2.0$ ): The experiments revealed that increasing the likelihood power beyond 1.0 (specifically to $\lambda=2.0$ ) minimizes the shot data residual, yielding the highest fidelity to the ground truth.
- Over-scaling ( $\lambda > 2.0$ ): Leads to overfitting and performance degradation.
- Key Insight: Maximum posterior fidelity does not necessarily occur at the classical Bayesian setting ( $\lambda=1$ ); up-weighting the likelihood can improve data alignment.
Joint Scaling (Power-Scaling Compass):
- High $\lambda$ + Low $\alpha$ : Strong adherence to observed RTM features but potentially less structural coherence.
- Low $\lambda$ + High $\alpha$ : Strong geological consistency but less responsiveness to specific data features.
- The method successfully generates diverse samples by tuning these parameters, allowing for exploration of the solution space.

5. Significance and Impact

Interpretability: The approach provides a principled mechanism to tune the "trust" placed in observed data versus prior geological knowledge, offering a new tool for sensitivity analysis in seismic inversion.
Computational Efficiency: By avoiding explicit likelihood evaluations during inference and using a single amortized network, the method achieves a generation time of approximately 4 seconds per sample, making it viable for practical applications.
Robustness: The ability to relax the prior (low $\alpha$ ) allows for the generation of diverse, exploratory models, which is crucial for uncertainty quantification in complex geological settings.
Generalizability: While demonstrated on seismic velocity models, the methodology is applicable to a broad class of inverse problems where balancing prior knowledge and data fidelity is critical.

In conclusion, this paper presents a significant advancement in Bayesian seismic inversion by integrating score-based generative models with power-scaling, offering a flexible, efficient, and interpretable framework for generating high-fidelity subsurface velocity models.