Positive-Negative Difference Adapter

• Positive-negative difference adapter is a neural module that models the discrepancy between normal and fault data via an additive perturbation approach.
• It integrates into a frozen diffusion-based backbone with local self-attention to capture fault-specific temporal patterns during few-shot adaptation.
• The training process employs combined denoising and diversity losses, leading to significant improvements in authenticity and diversity metrics.

A positive-negative difference adapter is a specialized neural module designed to model the domain shift between normal and fault distributions in few-shot fault time-series generation. It enables fine-grained adaptation of a diffusion-based generative model by leveraging a large-scale prior trained on abundant normal data while learning the minimal, targeted transformation necessary to synthesize authentic fault data under extreme scarcity. This approach is central to the FaultDiffusion framework for few-shot fault generation, enhancing both the authenticity and diversity of synthesized fault samples without catastrophic forgetting of normal domain representations (Xu et al., 19 Nov 2025).

1. Mathematical Formulation of Domain Shift

The positive-negative difference adapter is based on the observation that the distribution of fault data, denoted $p_f(x)$, can be modeled as a perturbation of the normal (pretrained) distribution, $p_n(x)$:

$p_f(x) = p_n(x) + \Delta_\theta(x)$

where $\Delta_\theta$ parameterizes the distributional difference needed to transform normal data patterns into fault patterns. This additive perturbation view enables targeted adaptation, focusing learning capacity on the domain discrepancies rather than relearning shared regularities (Xu et al., 19 Nov 2025).

2. Adapter Integration into the Diffusion Backbone

The implementation of the positive-negative difference adapter occurs at the level of frozen transformer layers within the diffusion backbone (specifically Diffusion-TS as in FaultDiffusion):

• The encoder and decoder networks are initialized and frozen after pretraining on abundant normal data.
• At each transformer layer $t$, a trained adapter operates on the local hidden representation. Let the backbone's hidden state at layer $t$ be $h_{\mathrm{bb}}^{(t)}$.
• The adapter receives aggregated hidden states from previous layers:

$$h_{\mathrm{in}}^{(t)} = h_{\mathrm{bb}}^{(t)} + \sum_{k=1}^{t-1} h_{\mathrm{loc}}^{(k)}$$

• The adapter applies a sliding-window multi-head self-attention mechanism, updating only a window of the sequence to capture local temporal fault structure:

$$x_{\mathrm{pad}} = \mathrm{Pad}(h_{\mathrm{in}}^{(t)}, \lfloor W/2 \rfloor)$$

$$h_{\mathrm{loc}}^{(t)} = \mathrm{MultiHeadAttn}\left(\mathrm{Window}(x_{\mathrm{pad}}, W)\right)_{\rm center}$$

• The output $h_{\mathrm{loc}}^{(t)}$ is residually fused with the backbone output at the subsequent layer:

$$h_{\mathrm{bb}}^{(t+1)} = h_{\mathrm{bb}}^{(t)} + \alpha h_{\mathrm{loc}}^{(t)} \;\; (\alpha=1)$$

• Only the adapter parameters $\phi$ are updated during few-shot fine-tuning; all other weights $\theta$ are frozen.

This construction allows the model to preserve rich normal-time-series representations while learning a compact, localized transformation for generating fault examples from scarce data (Xu et al., 19 Nov 2025).

3. Training Dynamics and Losses

The positive-negative difference adapter is optimized using a combination of standard denoising and explicit diversity losses:

• Denoising Loss: Standard DDPM objective encourages correct noise prediction for noisy inputs.

$$\mathcal{L}_{\rm base} = \mathbb{E}_{t,x_0,\epsilon}\left[\; \|\epsilon - \epsilon_\theta(x_t, t)\|_1 \;\right]$$

• Diversity Loss: Penalizes mode collapse by maximizing the squared difference between sampled outputs for the same input.

$$\mathcal{L}_{\rm div} = \mathbb{E}\left[\| s_1 - s_2 \|_2^2\right], \;\; s_1, s_2 \sim \epsilon_\theta(x_t, t)$$

• Total Objective:

$$\mathcal{L}_{\rm total} = \mathcal{L}_{\rm base} + \lambda \mathcal{L}_{\rm div}$$

where $\lambda$ balances fidelity and diversity.

The training involves (i) pretraining the backbone on normal data for up to 25,000 epochs, and (ii) fine-tuning only the adapter on $N_f$ scarce fault instances using $\mathcal{L}_{\rm total}$ for 5,000 updates, freezing all backbone weights. Generation proceeds by iterative denoising initialized from Gaussian noise (Xu et al., 19 Nov 2025).

4. Comparative Performance

The positive-negative difference adapter in FaultDiffusion demonstrates substantial improvements in authenticity and diversity of generated fault time series, as evaluated by metrics such as Context-FID, correlational score, discriminative score, and predictive score. In a head-to-head comparison on the Custom Industrial dataset (sequence length 24), the following table summarizes key quantitative results:

Method	Context-FID ↓	Corr-Score ↓	Disc-Score ↓	Pred-Score ↓
Cot-GAN	7.26	124.81	0.48	0.23
TimeGAN	8.17	139.05	0.49	0.24
TimeVAE	5.99	118.75	0.48	0.22
Diffusion-TS	6.73	109.30	0.50	0.23
Ours	6.08	106.61	0.42	0.13

Results on TEP and DAMADICS datasets confirm improvements in all metrics, with major gains in diversity as indicated by ACF-based diversity scores. Ablation studies show that removing either the adapter or the diversity loss degrades all metrics, confirming their necessity and complementarity (Xu et al., 19 Nov 2025).

5. Distinction from Other Few-Shot Time Series Adaptation Mechanisms

Alternative strategies for few-shot generative modeling in time series focus on domain-tuned conditional tokens (Gonen et al., 26 May 2025) or low-rank adapters for LLMs (Rousseau et al., 21 May 2025), but do not parameterize the explicit distributional shift as an additive difference to a fixed, pretrained normal prior. In contrast, the positive-negative difference adapter:

• Targets the minimal necessary parameter set (the adapter) for fast and robust adaptation.
• Operates within the frozen, structurally rich backbone to avoid overfitting and catastrophic forgetting, a documented issue with full network fine-tuning under data scarcity.
• Employs local self-attention in the adapter to capture fault-specific temporal patterns (Xu et al., 19 Nov 2025).

This suggests that difference-based adapter modules may be particularly effective in settings with large domain gaps and high intra-class variability, where parameter-efficient, locality-aware adaptation is required.

6. Limitations and Prospects for Future Work

Current limitations of the positive-negative difference adapter approach include its dependence on large-scale normal-data pretraining and the challenge of balancing adapter expressivity against the risk of overfitting to extremely limited fault data. The use of a fixed noise schedule may further restrict its flexibility across diverse fault types. Proposed directions to address these issues include meta-learning adapter initializations, introducing conditional guidance based on fault-specific embeddings, and developing diversity objectives aligned with higher-order statistics (Xu et al., 19 Nov 2025). Broader future trajectories involve applying similar adapter-based strategies to unified, cross-domain diffusion models and exploring their integration with LLM-based conditional generative frameworks (Gonen et al., 26 May 2025, Rousseau et al., 21 May 2025).