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Using Laplace Transform To Optimize the Hallucination of Generation Models

This paper proposes a novel framework that models generative systems as stochastic dynamical systems and utilizes Laplace transform analysis from a control theory perspective to fundamentally optimize and mitigate hallucinations in generation models.

Original authors: Cheng Kang, Xinye Chen, Daniel Novak, Xujing Yao

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cheng Kang, Xinye Chen, Daniel Novak, Xujing Yao

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

The Big Idea: Treating AI Like a Radio Tuner

Imagine you have a very talented but slightly crazy artist (the AI Model). This artist can paint beautiful pictures or write great stories, but sometimes they get too confident and start making things up that aren't true. In the AI world, this is called "Hallucination."

Usually, when people try to fix this, they look at the artist's brushstrokes (the code) or try to give them better reference books (more data).

This paper takes a completely different approach. The authors say, "Let's stop looking at the brushstrokes and start looking at the music." They treat the AI's learning process like a sound system or a radio. They use a mathematical tool called the Laplace Transform (think of it as a "super-spectrometer" for sound) to analyze how the AI "vibrates" while it learns.

The Problem: Too Much Vibration or No Vibration at All

The authors realized that for an AI to learn well, it needs the right amount of "vibration" or "shake."

Too much vibration: The AI gets jittery, overshoots the mark, and starts hallucinating (making up facts).
Too little vibration: The AI learns so slowly it never finishes, or it gets stuck in a rut.
Just right: The AI learns quickly, stays stable, and produces accurate results.

They found that the "vibration" of the AI depends heavily on the Optimizer it uses. An optimizer is like the engine or the fuel that pushes the AI to learn.

The Experiment: Testing Different Engines

The researchers tested several different "engines" (optimizers) on three types of AI artists:

GANs: Artists that create new images from scratch (like drawing a face from thin air).
CycleGANs: Artists that translate images (like turning a photo of a horse into a zebra).
DDPMs: Artists that clean up noisy images to reveal the picture underneath.

They used the "super-spectrometer" (Laplace Transform) to predict how each engine would make the AI vibrate, and then they actually ran the training to see if the prediction was right.

The Results: One Size Does Not Fit All

Here is what they discovered, using our analogies:

1. The "All-Rounder" Engine (Adam Optimizer)

Best for: Creating new images from scratch (GANs) and cleaning up noise (DDPM).
Why: This engine has a very strong, adaptive "shake." It can quickly adjust its rhythm to find the right note. If the AI starts to drift, this engine shakes it back into place fast.
The Analogy: It's like a sports car with a suspension system that adjusts itself instantly to bumps in the road. It prevents the car from crashing (hallucinating).

2. The "Precision" Engine (FuzzyPID Optimizer)

Best for: Translating images (CycleGAN).
Why: When turning a horse into a zebra, you need to keep the shape of the horse but change the stripes. The "FuzzyPID" engine is like a smart cruise control. It doesn't just shake; it listens to the road, remembers where it was a second ago, and predicts where it needs to go next. It keeps the translation smooth and prevents the AI from getting confused.
The Analogy: It's like a tightrope walker using a long pole to balance. If they lean too far left, the pole helps them correct it gently before they fall.

3. The "Old School" Engines (SGD, SGDM)

The Problem: These engines are too simple. They are like a car with no suspension. On a bumpy road (complex AI tasks), they bounce around too much or get stuck.
The Result: The AI either learns too slowly or starts hallucinating wildly because the engine can't handle the "noise" of the learning process.

The "Magic" Insight

The most exciting part of the paper is this: You don't need to wait for the AI to fail to know which engine to use.

By using the Laplace Transform, the researchers can simulate the AI's "heartbeat" on a computer before they even start training it.

If the simulation shows the AI will vibrate too wildly, they know to switch to a FuzzyPID engine.
If the simulation shows the AI is too sluggish, they know to switch to an Adam engine.

The Takeaway

Think of training an AI like tuning a radio.

Hallucination is just static noise.
Optimizers are the tuning knobs.
Laplace Transform is the visualizer that shows you exactly which knob to turn to get a clear signal.

The paper proves that by understanding the "physics" of how these models learn (using control theory), we can pick the perfect "tuning knob" for the job. This stops the AI from making things up and helps it produce reliable, high-quality results.

In short: Don't just guess which tool to use for your AI. Analyze its "vibration" first, pick the right engine, and you'll get a masterpiece instead of a hallucination.

1. Problem Statement

Generation Models (GMs), including Large Language Models (LLMs), Vision-Language Models, and Generative Adversarial Networks (GANs), suffer from hallucinations (confident errors where models generate factually incorrect or out-of-distribution content).

Current Limitations: Existing research primarily addresses hallucinations at a "microscopic" level by modifying model architectures, adding retrieval mechanisms (RAG), or tweaking loss functions.
The Gap: There is a lack of research analyzing the stability and convergence of the learning systems themselves from a "macroscopic" perspective. The fundamental reason why GMs produce confident errors remains unclear, and there is no systematic framework to determine which optimization methods best suit specific generation tasks.

2. Methodology

The authors propose a novel framework that formalizes Generation Models as stochastic dynamical systems using Control Theory. The core methodology involves:

System Formalization: The training process of GMs is modeled as a control system where the parameter update ( $\theta$ ) is the system output and the gradient is the error signal.
Laplace Transform Analysis: The authors apply the Laplace transform to derive the system functions ( $\theta(s)$ ) of various optimization algorithms. This allows them to analyze stability, convergence, and frequency response without needing to solve the complex non-linear equations of the actual neural networks.
Optimizer Modeling: Several optimizers are mapped to control theory components:
- SGD: Modeled as a P-controller (Proportional).
- SGDM: Modeled as a PI-controller (Proportional-Integral).
- PID Optimizer: Modeled as a full PID-controller.
- Adam: Analyzed as a complex adaptive system (though its exact system function is too complex to derive analytically, its behavior is simulated).
- Filter-based SGD: Modeled using transfer functions for Low-Pass (LPF) and High-Pass (HPF) filters.
- FuzzyPID: A PID controller where gains are adjusted via fuzzy logic to handle non-linear variations.
Simulation: Using Simulink (MATLAB), the authors simulate the system responses of these optimizers when applied to three specific GM architectures:
1. Classical GAN (Single generator/discriminator).
2. CycleGAN (Two generators/discriminators with cycle consistency).
3. DDPM (Denoising Diffusion Probabilistic Models).
Experimental Validation: The theoretical simulations are validated by training these models on MNIST and USPS datasets using PyTorch, comparing the generated samples against the simulated system responses.

3. Key Contributions

Theoretical Framework: Established a connection between Control Theory and Deep Learning, formalizing GM hallucinations as issues of system instability and vibration (overshoot) in the learning process.
System Function Derivation: Successfully derived the Laplace domain system functions for standard optimizers (SGD, SGDM, PID) and analyzed their stability poles.
Optimization-System Matching: Demonstrated that different GM architectures have distinct "system responses" and require specific optimizers to ensure stability:
- GANs require high-frequency adaptability (Adam).
- CycleGANs require stability and cycle consistency (PID/FuzzyPID).
- DDPMs require adaptive vibration to prevent divergence (Adam-family).
Hallucination Mitigation Strategy: Proposed that hallucinations can be fundamentally reduced by selecting an optimizer that matches the system's dynamic requirements, ensuring the training process converges without excessive oscillation or drift.

4. Results and Analysis

The study found a high consistency between the simulated system responses and the actual generated samples:

Classical GAN:
- Simulation: SGD, SGDM, and PID showed marginal stability or baseline drift. Adam showed rapid convergence and stability.
- Result: Adam generated clear, diverse samples. Other optimizers produced duplicate samples or high-noise outputs (hallucinations) due to divergence.
CycleGAN:
- Simulation: FuzzyPID and PID provided the most stable sinusoidal responses with minimal drift. SGD and Adam showed divergence risks.
- Result: FuzzyPID produced "flawless" samples from the outset and maintained perfect cycle consistency. SGD and Adam failed to generate recognizable images after 100 epochs.
DDPM:
- Result: Only Adam-family optimizers (Adam, AdamW, RAdam, NAdam) produced readable samples. Non-adaptive methods (SGD, PID) failed to converge, resulting in noisy, hallucinated outputs.
Mechanism of Failure: The paper attributes hallucinations to overshoot and vibration in the training process.
- Too little vibration (e.g., standard SGD) leads to slow learning or getting stuck.
- Too much vibration or lack of adaptive control leads to divergence and mode collapse.
- Adam succeeds in GANs/DDPM because its adaptive nature provides the necessary "vibration" to adjust quickly.
- FuzzyPID succeeds in CycleGAN because it stabilizes the complex dual-generator system without the aggressive oscillation that causes divergence in cycle-consistency tasks.

5. Significance

Paradigm Shift: Moves the focus of hallucination research from data augmentation and loss function tweaking to system stability analysis.
Predictive Power: The Laplace transform approach allows researchers to predict the performance and stability of an optimizer on a specific GM architecture before training, saving computational resources.
Design Guidelines: Provides a concrete rule: "Match the optimizer to the system."
- Use Adaptive/High-Vibration optimizers (Adam) for single-generator, noise-to-sample tasks (GANs, DDPM).
- Use Stable/Controlled optimizers (PID/FuzzyPID) for multi-generator, constraint-heavy tasks (CycleGAN).
Future Impact: Suggests that control theory principles can guide the development of new optimization components for all types of Artificial Neural Networks (ANNs) to fundamentally reduce confident errors and improve reliability.