Learnable Physics (Physics-Informed ML) is the interdisciplinary field at the intersection of deep learning and scientific computing that combines data-driven neural network learning with known physical laws (conservation principles, governing PDEs, symmetries) to create models that are both flexible enough to learn from data and constrained enough to respect fundamental physics — addressing the critical limitation that pure data-driven models can produce physically impossible predictions while pure physics simulations cannot adapt to real-world complexity beyond their governing equations.

What Is Learnable Physics?

• Definition: Learnable physics encompasses any approach that integrates domain knowledge from physics into machine learning models — either as soft constraints (physics-based loss terms), hard constraints (architecture design), training data augmentation (physics simulation for data generation), or hybrid systems (neural networks correcting physics simulators).
• The Spectrum: At one end, Physics-Informed Neural Networks (PINNs) learn to solve specific PDEs by penalizing violations of the governing equation in the loss function. At the other end, Neural Operators (Fourier Neural Operator, DeepONet) learn the entire solution operator — mapping from boundary/initial conditions to solutions — potentially replacing traditional PDE solvers entirely.
• Data Efficiency: Pure data-driven models require enormous training datasets because they must learn both the underlying physics and the specific solution simultaneously. Physics-informed approaches embed the physics as prior knowledge, dramatically reducing the data needed to learn accurate solutions — often achieving good accuracy from sparse, noisy observations.

Why Learnable Physics Matters

• Physical Validity: Standard neural networks can predict negative energies, superluminal velocities, or mass-violating trajectories because they have no knowledge of conservation laws. Physics-informed models enforce these constraints, producing predictions that scientists can trust for engineering decisions.
• Inverse Problem Solving: Many scientific problems are inverse — "given observations, what are the governing parameters?" PINNs naturally solve inverse problems by treating unknown parameters as learnable variables optimized alongside the neural network weights, simultaneously fitting the data and the physics.
• Speed vs. Accuracy: Traditional PDE solvers (finite element, finite difference) are accurate but computationally expensive — a single CFD simulation can take hours or days. Trained neural surrogates produce approximate solutions in milliseconds, enabling real-time design optimization, uncertainty quantification, and interactive exploration of parameter spaces.
• Beyond Governing Equations: Many real-world systems have partially known physics — the governing equations capture the dominant behavior but miss secondary effects (turbulence closure, sub-grid phenomena, constitutive relations). Neural networks can learn these missing components from data while the known physics provides the structural backbone.

Physics-Informed ML Approaches

Learnable Physics is guided discovery — using deep learning to solve scientific problems while forcing the model to obey the conservation laws, symmetries, and governing equations that nature enforces, producing AI systems that a physicist can trust.