发表机构
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine; Universidad Politécnica de Cartagena; Gebze Technical University; Nicolaus Copernicus University in Toruń（乌克兰国家科学院B.维尔金低温物理与工程研究所; 卡塔赫纳理工大学; 格布泽理工大学; 托伦哥白尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出距离残差物理信息神经网络（DR-PINNs），用距离平方替代点残差求解微分包含，证明相容性，并在基准上实现高精度。
AI 中文摘要
我们提出了距离残差物理信息神经网络（DR-PINNs），这是一种物理信息学习框架，用于逼近常微分和偏微分包含（DIs）的解，这些包含是微分算子被约束在集值映射中而非等于指定函数的控制定律。该方法将经典的逐点PDE/ODE残差替换为微分算子到允许集的距离的平方。当包含被满足时，该距离恰好为零，并度量算子进入允许集所需的最小修正。对于固定的闭凸允许集，平方距离关于算子值是可微的，其梯度由度量投影给出。当允许集还依赖于网络状态时，该依赖性通过链式法则包含在内。该框架涵盖具有集值反应项的常微分和偏微分包含。对于这两种情况，我们证明了相容性：在所述的闭性、可测性、凸性和增长假设下，任何满足初始（在抛物型情形下，边界）条件的候选序列，如果其连续距离残差泛函趋于零，则存在一个子序列收敛到目标包含的精确解。这些是针对连续距离残差泛函的条件性陈述；它们不涵盖有限配点训练损失、优化器的行为或收敛速率。对于允许集为有限多个顶点的凸包的情况，到该集合的投影简化为一个小型凸二次规划，使得损失在训练循环内可高效计算。数值实验在微分包含基准上展示了高精度。
英文摘要
We introduce Distance-Residual Physics-Informed Neural Networks (DR-PINNs), a physics-informed learning framework for approximating solutions of ordinary and partial differential inclusions (DIs), governing laws in which a differential operator is constrained to lie in a set-valued map rather than equaling a prescribed function. The method replaces the classical pointwise PDE/ODE residual by the squared distance from the differential operator to the admissible set. This distance vanishes exactly when the inclusion is satisfied and measures the infimal correction needed for the operator to enter the admissible set. For a fixed closed convex admissible set, the squared distance is differentiable with respect to the operator value, with gradient given by the metric projection. When the admissible set also depends on the network state, that dependence is included through the chain rule. The framework encompasses ordinary and partial DIs with set-valued reaction terms. For both settings we prove consistency: under the stated closedness, measurability, convexity, and growth assumptions, any sequence of candidates satisfying the initial (and, in the parabolic case, boundary) conditions whose continuous distance-residual functional tends to zero admits a subsequence converging to an exact solution of the target inclusion. These are conditional statements for the continuous distance-residual functional; they do not cover the finite-collocation training loss, the behavior of the optimizer, or convergence rates. For admissible sets given as convex hulls of finitely many vertices, projection onto the set reduces to a small convex quadratic program, making the loss efficiently computable inside the training loop. Numerical experiments demonstrate high accuracy on the differential-inclusion benchmarks.
