In this paper, we propose a new method called Multi-Modal Gradual Domain Osmosis, which aims to solve the problem of smooth knowledge migration from the source domain to the target domain in Gradual Domain Adaptation (GDA). Traditional Gradual Domain Adaptation methods mitigate domain bias by introducing intermediate domains and self-training strategies but often face the challenges of inefficient knowledge migration or missing data in intermediate domains. In this paper, we design an optimization framework based on the hyperparameter łambda by dynamically balancing the loss weights of the source and target domains, which enables the model to progressively adjust the strength of knowledge migration (łambda incrementing from 0 to 1) during the training process, thus achieving cross-domain generalization more efficiently. Specifically, the method incorporates self-training to generate pseudo-labels and iteratively updates the model by minimizing a weighted loss function to ensure stability and robustness during progressive adaptation in the intermediate domain. The experimental part validates the effectiveness of the method on rotated MNIST, color-shifted MNIST, portrait dataset, and forest cover type dataset, and the results show that it outperforms existing baseline methods. The paper further analyses the impact of the dynamic tuning strategy of the hyperparameter łambda on the performance through ablation experiments, confirming the advantages of progressive domain penetration in mitigating domain bias and enhancing the model generalization capability. The study provides theoretical support and a practical framework for asymptotic domain adaptation and expands its application potential in dynamic environments.

In this paper, we propose a new method called \textit{Self-Training with Dynamic Weighting} (STDW), which aims to enhance robustness in Gradual Domain Adaptation (GDA) by addressing the challenge of smooth knowledge migration from the source to the target domain. Traditional GDA methods mitigate domain shift through intermediate domains and self-training but often suffer from inefficient knowledge migration or incomplete intermediate data. Our approach introduces a dynamic weighting mechanism that adaptively balances the loss contributions of the source and target domains during training. Specifically, we design an optimization framework governed by a time-varying hyperparameter $\varrho$ (progressing from 0 to 1), which controls the strength of domain-specific learning and ensures stable adaptation. The method leverages self-training to generate pseudo-labels and optimizes a weighted objective function for iterative model updates, maintaining robustness across intermediate domains. Experiments on rotated MNIST, color-shifted MNIST, portrait datasets, and the Cover Type dataset demonstrate that STDW outperforms existing baselines. Ablation studies further validate the critical role of $\varrho$'s dynamic scheduling in achieving progressive adaptation, confirming its effectiveness in reducing domain bias and improving generalization. This work provides both theoretical insights and a practical framework for robust gradual domain adaptation, with potential applications in dynamic real-world scenarios.