Introductory Notes on Learning2
Abstract
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 Learning2, 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 Learning2 through EuLaNet, an Eulerian--Lagrangian representation for fluid dynamics. Given the velocity state u(x,t), EuLaNet constructs its induced Lagrangian flow map X(a,t) through Ẋ(a,t)=u(X(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 ℋL2⊆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 Learning2 architectures. The implementation is open-sourced to support the development and extension of the architecture across scientific domains.
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BibTeX
@misc{siddharth2026introductory,
title = {{Introductory Notes on Learning$^2$}},
author = {Sai Siddharth and Maniarasu Ravi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06546},
eprint = {2609.06546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06546},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Introductory Notes on Learning^2
AU - Sai Siddharth
AU - Maniarasu Ravi
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06546
UR - https://arxiv.org/abs/2609.06546
AB - 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(x,t), EuLaNet constructs its induced Lagrangian flow map X(a,t) through Ẋ(a,t)=u(X(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 ℋ_L^2⊆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.
ER -