AI 中文总结
研究刚性和冲击主导型偏微分方程求解问题，提出SPARC-Net架构及训练框架，通过自适应多尺度频谱编码器等解决相关问题，经多基准验证比普通PINNs有显著改进，如降低各方程误差，还进行多项分析。
AI 中文摘要
物理信息神经网络（PINNs）为求解偏微分方程（PDEs）提供了一种无网格方法，但在刚性和冲击主导问题中会严重退化，小的PDE残差可能对应全局不准确的解。我们表明这些失败是多因素导致的，源于（i）对尖锐特征的频谱偏差、（ii）不平衡的多项优化和损失权重崩溃、（iii）违反时间因果关系以及（iv）配置点解析不足。我们提出了SPARC-Net，一个统一的架构和训练框架，共同解决这四个问题。SPARC-Net利用具有可学习频谱门的自适应多尺度频谱编码器、门控残差主干、自适应激活以及精确执行初始和边界条件的硬约束输出假设，从结构上消除损失权重崩溃。训练采用稳定的梯度范数损失平衡、尊重因果关系的下限残差加权以及基于残差的自适应配置（RAD）。通过四个标准基准（粘性伯格斯方程、艾伦-卡恩方程、对流（β = 30）和反应）的精确解析和高阶频谱参考解验证，SPARC-Net比普通PINNs有显著改进：伯格斯方程的相对L2误差从1.47e-1降至1.14e-1（降低22%），艾伦-卡恩方程从9.93e-1降至5.78e-2（降低94%），反应方程从9.82e-1降至3.54e-3（降低100%）。双曲传输的特征坐标编码器进一步将对流误差从5.14e-1降至9.88e-5（降低100%）。我们报告了五种子均值±标准差误差、威尔科克森显著性检验、全面的消融研究、超参数敏感性、二维热方程的扩展以及与参数匹配基线的比较。
英文摘要
Physics-Informed Neural Networks (PINNs) provide a meshless approach for solving partial differential equations (PDEs), but suffer severe degradation in stiff and shock-dominated problems, where small PDE residuals can correspond to globally inaccurate solutions. We show these failures are multi-causal, arising from the concurrent interplay of (i) spectral bias against sharp features, (ii) imbalanced multi-term optimization and loss-weight collapse, (iii) violation of temporal causality, and (iv) under-resolved collocation. We present SPARC-Net, a unified architecture and training framework that jointly addresses all four pathologies. SPARC-Net leverages an adaptive multi-scale spectral encoder with a learnable spectral gate, a gated residual backbone, adaptive activations, and a hard-constraint output ansatz that exactly enforces initial and boundary conditions, structurally eliminating loss-weight collapse. Training employs stabilized gradient-norm loss balancing, floored causality-respecting residual weighting, and residual-based adaptive collocation (RAD). Validated against exact analytic and high-order spectral reference solutions across four canonical benchmarks -- viscous Burgers', Allen-Cahn, convection (beta=30), and reaction -- SPARC-Net yields substantial improvements over vanilla PINNs: relative L2 error drops from 1.47e-1 to 1.14e-1 on Burgers' (22% reduction), 9.93e-1 to 5.78e-2 on Allen-Cahn (94% reduction), and 9.82e-1 to 3.54e-3 on reaction (100% reduction). A characteristic-coordinate encoder for hyperbolic transport further reduces convection error from 5.14e-1 to 9.88e-5 (100% reduction). We report five-seed mean +/- standard deviation errors, Wilcoxon significance tests, full ablation studies, hyperparameter sensitivities, an extension to the 2D heat equation, and comparisons against parameter-matched baselines.
Comments8 pages, 11 figures, 5 tables
