## 2 Proposed Method
### Adversarial Task Augmented Sequential Meta-Learning (ATASML) Framework
#### 2.3.2 Design of Adversarial Samples  
Adversarial samples within the ATASML framework are specifically designed to evaluate and enhance the model's resilience against operational conditions that mimic real-world disturbances. These samples are derived from both the augmented dataset \(\mathcal{X}_{aug}\), which comprises samples modified by standard data-augmentation techniques, and the original dataset \(\mathcal{X}_{orig}\).  
The generation of adversarial samples, depicted in Figure 2, involves the following steps:  
1. Dataset Composition: The dataset for generating adversarial samples, \(\mathcal{X}_{adv}\), is formulated using: * \(\mathcal{X}_{gauss}\): Samples augmented with Gaussian noise. * \(\mathcal{X}_{uarp}\): Samples modified through temporal warping. These augmented samples, along with unaltered original samples from \(\mathcal{X}_{orig}\), are used to construct \(\mathcal{X}_{adv}\).
2. Strategic Sample Selection (\(S\)): A subset \(S=\{x^{\prime}_{i}\}\) from both \(\mathcal{X}_{aug}\) (comprising \(\mathcal{X}_{gauss}\) and \(\mathcal{X}_{uarp}\)) and \(\mathcal{X}_{orig}\) is selected to create instances that exhibit potential vulnerabilities under varied operational conditions. This targeted selection process is designed to challenge the model realistically and robustly: * Focusing on samples that represent critical transitional states or dynamic conditions. * Electing samples that simulate rare operational disruptions or extreme conditions.  
Figure 2: Schematic representation of adversarial sample generation in the ATASML framework. The process uses augmented samples (\(\mathcal{X}_{gauss}\) and \(\mathcal{X}_{uarp}\)) combined with the original samples (\(\mathcal{X}_{orig}\)) to create a comprehensive set of adversarial samples (\(\mathcal{X}_{adv}\)).  
* Choosing instances that might indicate potential system failures or significant performance anomalies.
* Adversarial Optimization Problem: The adversarial samples \(\mathcal{X}_{adv}\) are generated by solving an optimization problem formulated to maximize the predictive error, thereby testing the model's robustness: \[\mathcal{X}_{adv}=\left\{x_{adv_{i}}\mid x_{adv_{i}}=\arg\max_{x_{i}^{\prime}} \mathcal{L}(f(x_{i}^{\prime}),y_{i}^{\prime}),\text{s.t.}\|\hat{x}_{i}^{\prime }-x_{i}^{\prime}\|_{p}\leq\epsilon\right\}\] (5) where \(\hat{x}_{i}^{\prime}=x_{i}^{\prime}+\delta\), \(\delta=\epsilon\cdot\text{sign}(\
abla_{x_{i}^{\prime}}\mathcal{L}(f(x_{i}^{ \prime}),y_{i}^{\prime}))\) is the perturbation designed to maximally disrupt the model's predictions, and \(y_{i}^{\prime}\) represents the labels corresponding to the strategically selected samples, denoting their operational states as normal or anomalous.
* Gradient Sign Method: This method employs the gradient of the loss function with respect to the inputs from \(S\) to determine the most effective direction for perturbations, thereby ensuring the adversarial samples are optimally challenging: \[\text{sign}(\
abla_{x_{i}^{\prime}}\mathcal{L})=\text{sign}(\
abla_{x_{i}^{ \prime}}\mathcal{L}(f(x_{i}^{\prime}),y_{i}^{\prime}))\] (6) This systematic approach of employing both \(\mathcal{X}_{aug}\) (including \(\mathcal{X}_{gauss}\) and \(\mathcal{X}_{warp}\)) and \(\mathcal{X}_{orig}\) to generate \(\mathcal{X}_{adv}\) ensures the model is comprehensively evaluated against synthetic distortions as well as baseline conditions. Such rigorous testing is essential for verifying the model's operational reliability across diverse and potentially disruptive conditions.