针对物理信息神经网络任务差异大的问题,提出模块化元学习框架,提升跨任务泛化能力。
Compositional Meta-Learning for Mitigating Task Heterogeneity in Physics-Informed Neural Networks
- 基于任务学习动态构建表征,用聚类划分子网络,实现任务专用模块复用。
- 在未见任务上比传统PINN平均降低19.7倍均方误差,仅需10%训练迭代。
- 适合资源受限工程场景中参数化偏微分方程的快速部署与泛化。
物理信息神经网络(PINNs)通过将物理定律嵌入损失函数来近似偏微分方程(PDE)的解。在参数化PDE族中,系数或边界/初始条件的变化定义了不同任务。为每个任务单独训练PINNs计算成本过高,而跨任务迁移易受任务异质性影响。尽管元学习可降低重训练开销,但现有方法常依赖单一全局初始化,可能产生负向迁移,尤其在坐标输入特征稀缺且训练任务有限时。本文提出学习亲和力自适应模块化物理信息神经网络(LAM-PINN),一种组合式框架,利用任务特异性学习动态。LAM-PINN结合PDE参数与简短迁移会话中的学习亲和度度量,构建任务表示并实现聚类,即使仅有坐标输入也可行。它将模型分解为簇专用子网络与共享元网络,并学习路由权重以选择性复用模块,而非依赖单一全局初始化。在三个PDE基准测试中,LAM-PINN在未见任务上平均实现19.7倍的均方误差(MSE)降低,仅需传统PINNs 10%的训练迭代。结果表明,该方法在参数化PDE族的有限设计空间内,对未见配置具有优异泛化能力,适用于资源受限的工程场景。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) approximate solutions of partial differential equations (PDEs) by embedding physical laws into the loss function. In parameterized PDE families, variations in coefficients or boundary/initial conditions define distinct tasks. This makes training individual PINNs for each task computationally prohibitive, while cross-task transfer can be sensitive to task heterogeneity. While meta-learning can reduce retraining cost, existing methods often rely on a single global initialization and may suffer from negative transfer, particularly under feature-scarce coordinate inputs and limited training-task availability. We propose the Learning-Affinity Adaptive Modular Physics-Informed Neural Network (LAM-PINN), a compositional framework that leverages task-specific learning dynamics. LAM-PINN combines PDE parameters with learning-affinity metrics from brief transfer sessions to construct a task representation and cluster tasks even with coordinate-only inputs. It decomposes the model into cluster-specialized subnetworks and a shared meta network, and learns routing weights to selectively reuse modules instead of relying on a single global initialization. Across three PDE benchmarks, LAM-PINN achieves an average 19.7-fold reduction in mean squared error (MSE) on unseen tasks using only 10% of the training iterations required by conventional PINNs. These results indicate its effectiveness for generalization to unseen configurations within bounded design spaces of parameterized PDE families in resource-constrained engineering settings.
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