Using Laplace Transform To Optimize the Hallucination of Generation Models

Cheng Kang; Xinye Chen; Daniel Novak; Xujing Yao

math.OCMar 5, 2026

Using Laplace Transform To Optimize the Hallucination of Generation Models

Authors: Cheng Kang, Xinye Chen, Daniel Novak, Xujing Yao

Organizations: 1Cheng Kang and Daniel Novak are with the Department of Cybernetics, Czech Technical University in Prague, 16000 Praha 6, Prague, Czech Republic · 2Xinye Chen is with Department of Numerical Mathematics, Charles University, 11000 Praha 1, Prague, Czech Republic · 3Xujing Yao is with the College of Computer and Information Engineering, Nanjing Tech University, Nanjing, Jiangsu 211816, China

Abstract

To explore the feasibility of avoiding the confident error (or hallucination) of generation models (GMs), we formalise the system of GMs as a class of stochastic dynamical systems through the lens of control theory. Numerous factors can be attributed to the hallucination of the learning process of GMs, utilising knowledge of control theory allows us to analyse their system functions and system responses. Due to the high complexity of GMs when using various optimization methods, we cannot figure out their solution of Laplace transform, but from a macroscopic perspective, simulating the source response provides a virtual way to address the hallucination of GMs. We also find that the training progress is consistent with the corresponding system response, which offers us a useful way to develop a better optimization component. Finally, the hallucination problem of GMs is fundamentally optimized by using Laplace transform analysis.

Figures & tables

Figure 1: Evolution process of the weight $\theta({t})$ for each component under the generation task.

Figure 2: The control system of classical GAN.

Panel of Figure 3 — Figure 3: The system response of Classical GAN on various optimizers. Optimizer from left to right is respectively SGD, SGDM, Adam, PID, LPFSGD, HPFSGD, and FuzzyPID. The input of generator is a random Gaussian noise, and the input of discriminator is a square wave. (Blue wave comes from the discriminator, and yellow wave is the response of the generator.)

Figure 4: The control system of CycleGAN.

Panel of Figure 5 — Figure 5: The system response of CycleGAN on different optimizers, such as SGD, SGDM, Adam, PID, LPF-SGD, HPF-SGD and FuzzyPID optimizers. For $G_{a}$ , the source is a sinusoidal wave (green), and the dashed green wave comes from $D_{a}$ . For $G_{a}$ , the source is a cosine wave (red), and the dashed red wave comes from $D_{b}$ .

Hyper-parameter	GAN	CycleGAN	DDPM
Data Name	MNIST	MNIST and UPSP	MNIST
Data augmentation	Auto	Auto	Auto
Input resolution	[28,28,1]	[28,28,1]	[28,28,1]
Epochs	200	200	100
Batch size	200	200	200
Hidden dropout	0	0	0

Table I : Hyper-parameters for the image generation task on MNIST and UPSP.

Filter Type	Gain	Numerator			Denominator
Filter Type	$G$	$x_{0}$	$x_{1}$	$x_{2}$	$y_{0}$	$y_{1}$	$y_{2}$
Low Pass Filter	0.49968	1	-0.99937	0.00063	1.0	0	-1.0
High Pass Filter	0.49968	1	0.99937	0.00063	1.0	0	-1.0

Table II : Coefficients of LPF-SGD and HPF-SGD using second-order IIR structure.

Figure 6: The generated samples from classical GAN on corresponding optimizers (from top to bottom is respectively SGD, SGDM, Adam, PID, LPF-SGD, HPF-SGD, and FuzzyPID).

Figure 7: The generated samples from CycleGAN on corresponding optimizers (From top to bottom is SGD, SGDM, Adam, PID, LPF-SGD, HPF-SGD and FuzzyPID). The generated confident errors are marked with a red line.

Panel of Figure 8 — Figure 8: The generated samples from DDPM on corresponding optimizers (From top to bottom is SGD, SGDM, Adam, PID, LPF-SGD, HPF-SGD and FuzzyPID).

Panel of Figure 9 — Figure 9: The generated samples from DDPM on corresponding optimizers (From top to bottom is SGD, SGDM, Adam, PID, LPF-SGD, HPF-SGD and FuzzyPID).

Panel of Figure 10 — Figure 10: The generated samples from DDPM on corresponding optimizers (From top to bottom is SGD, SGDM, Adam, PID, LPF-SGD, HPF-SGD and FuzzyPID).

Hallucination Mitigation

Hallucination mitigation develops methods to detect and reduce unsupported AI outputs while preserving useful, evidence-grounded responses.

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Recent work on hallucination detection in large language models has shown that, for a fixed pre-trained model and reasoning task, it is possible to estimate the model's confidence in the correctness of its outputs. Such uncertainty estimates have primarily been used to improve truthfulness by detecting or filtering confabulations. In this work, we ask whether these signals can instead be used more proactively to directly improve the accuracy of model-generated answers. We propose USteer, a simple, training-free steering mechanism that adjusts a model's layer-wise activations during inference using the gradient of a confidence measure with respect to the activations. This procedure nudges generation toward outputs with lower uncertainty at inference time, without modifying model parameters or requiring additional supervision. We show that this approach consistently reduces hallucination across a range of tasks, demonstrating that confidence signals can be leveraged not only for detection, but also for effective inference-time control of model behavior.

Abstract

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