发表机构
Thiagarajar College of Engineering; Massachusetts Institute of Technology（蒂亚加拉贾尔工程学院; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出Learning$^2$框架，通过物理变换耦合主表示与第二表示以约束假设空间，并基于欧拉-拉格朗日表示实现EuLaNet，为科学学习提供事前物理可解释约束。
AI 中文摘要
尽管机器学习可用于从数据预测物理系统的演化，但仅学习每个时刻系统状态的公式化方法，将解的时间与动力学结构留待在一个宽泛的假设空间内求解。我们引入Learning$^2$，一个表示层框架，通过将主表示与第二表示经由已知物理变换耦合来结构化该空间。由此产生的跨表示约束限制了有效的假设空间，并提供了一种事前（ante-hoc）、物理上可解释的准则，用于排除仅满足主表示的候选解。我们通过EuLaNet（一种用于流体动力学的欧拉-拉格朗日表示）实例化Learning$^2$。给定速度状态$u(\mathbf{x},t)$，EuLaNet通过$\dot{X}(\mathbf{a},t)=u(X(\mathbf{a},t),t)$构造其诱导的拉格朗日流映射$X(\mathbf{a},t)$，并由此导出物质输运与有限时间变形。所得表示将预测状态与其诱导的动力学后果耦合，提供了超越状态级一致性的第二一致性准则。我们通过有效假设空间$\mathcal{H}_{L^2}\subseteq\mathcal{H}$形式化该构造，并定义了后果表示为候选解提供判别性约束的条件。EuLaNet作为模型无关的表示模块实现，将物理约束与下游学习架构分离。该构造为科学学习中的物理可解释约束提供了一种事前机制，并为开发和评估更广泛的Learning$^2$架构提供了基础。该实现已开源，以支持该架构在科学领域中的开发与扩展。
英文摘要
Although machine learning can be used to predict the evolution of physical systems from data, a formulation that learns only the system state at each time leaves the temporal and dynamical structure of the solution to be resolved within a broad hypothesis space. We introduce Learning$^2$, a representation-level framework that structures this space by coupling a primary representation to a second representation through a known physical transformation. The resulting cross-representation constraint restricts the effective hypothesis space and provides an ante-hoc, physically interpretable criterion for excluding solutions that satisfy the primary representation alone. We instantiate Learning$^2$ through EuLaNet, an Eulerian--Lagrangian representation for fluid dynamics. Given the velocity state $u(\mathbf{x},t)$, EuLaNet constructs its induced Lagrangian flow map $X(\mathbf{a},t)$ through $\dot{X}(\mathbf{a},t)=u(X(\mathbf{a},t),t)$, from which material transport and finite-time deformation are derived. The resulting representation couples the predicted state to the dynamical consequences it induces, providing a second consistency criterion beyond state-level agreement. We formalize this construction through an effective hypothesis space $\mathcal{H}_{L^2}\subseteq\mathcal{H}$ and define the conditions under which a consequence representation provides discriminative constraints on candidate solutions. EuLaNet is implemented as a model-independent representation module, separating the physical constraint from the downstream learning architecture. This construction provides an ante-hoc mechanism for physically interpretable constraint in scientific learning and offers a basis for developing and evaluating broader classes of Learning$^2$ architectures. The implementation is open-sourced to support the development and extension of the architecture across scientific domains.
Comments11 pages. Code and implementation: https://github.com/EuLaNet/EuLaNet
