傳統的多層感知器（MLP）傾向於平滑插值，因此在學習具有尖銳、高對比轉換的函數時往往效果不佳。本論文提出輕量級架構 Level-Set Neural Decomposition（LSND），將不連續性的偵測與局部函數近似解耦。首先，淺層網路回歸一個平滑的level-set場，其零等值線可柔性地將定義域分割；之後透過可微的二階閘控算子將區域專屬的單層感知器進行混合，使模型能在學習到的介面上放置任意陡峭的跳躍，同時保留梯度式訓練所需的可微性。
為了初始化介面，我們採用基於斜率的啟發式方法，在梯度幅值較大處自動細分取樣密度，無需耗費大量搜尋或重度正則化即可獲得精確的跳躍候選點。在一維組合訊號及二維含花瓣形、雙圓形不連續面的迴歸任務中，LSND相較於寬度匹配的MLP，可將均方誤差降低一到兩個數量級，同時參數量減少10–30％、訓練時間約為一半。此優勢在複雜介面附近最為顯著：level-set 分支加速收斂，而域分解子網路避免了單一模型常見的假振盪。
除實驗成效外，LSND亦帶來概念層面的啟示：透過於網路中嵌入幾何先驗，可將深度學習的表現力與分段光滑函數的結構相結合，為科學計算、電腦圖學與訊號處理中的不連續感知網路提供一條通用思路。目前的限制包括僅考慮單一、非交錯介面且跳躍方向一致；未來可望擴展至分支或高度震盪的不連續情形

Learning functions that exhibit sharp, high-contrast transitions remains a challenge for conventional multilayer perceptrons (MLPs), which are biased toward smooth interpolation. This thesis introduces Level-Set Neural Decomposition (LSND), a lightweight architecture that decouples discontinuity detection from local function approximation. A shallow network first regresses a smooth level-set field whose zero-contour softly partitions the domain; region-specific single-layer perceptrons are then blended through a differentiable second-order gating operator, allowing the model to place an arbitrarily steep jump at the learned interface while preserving differentiability for gradient-based training.
To initialize the interface, we employ a slope-based heuristic that iteratively refines sample density where gradient magnitudes are large, yielding accurate jump candidates without exhaustive search or heavy regularization.Comprehensive experiments on one-dimensional composite signals and two-dimensional regression tasks with petal-shaped and dual-circle discontinuities demonstrate that LSND attains one to two orders of magnitude lower mean-squared error than width-matched MLPs, while using 10–30% fewer parameters and roughly half the training time.These gains are most pronounced near complex interfaces, where the level-set branch expedites convergence and the domain-decomposed subnetworks avoid the spurious oscillations typical of monolithic models.
Beyond its empirical advantages, LSND offers conceptual insight: by embedding geometric priors directly into the network, we reconcile the expressiveness of deep learning with the structure of piecewise-smooth functions, suggesting a general recipe for discontinuity-aware architectures in scientific computing, computer graphics, and signal processing. Current limitations include the assumption of a single, non-intersecting interface and uniform jump direction; extending LSND to handle branching or highly oscillatory discontinuities constitutes promising future work.

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