&Bullet; physics 14, s25

A neural network can be made to produce more reliable predictions of nonlinear systems when it is made with built-in conservation theorems.

R. Gauthier-Butterfield / UC Irvine

A baby learns how objects move by observing them. But without considering the conservation of momentum, his developed understanding is just an educated guess. Similarly, an artificial neural network (ANN) learns from empirical data how a particular system works, but without explicitly considering the conservation laws governing that system, it risks making unreliable predictions. To overcome this limitation, Tom Beucler of the University of California, Irvine, and colleagues devised a way to hardwire an ANN with such laws. They demonstrated the technique using an atmospheric climate model, but they say their method can be applied to models of any physical system [1] .

An KNN is an algorithm-based tool for converting a set of inputs into a set of outputs. The development or “training” process for an ANN involves incrementally reconfiguring its underlying algorithm with data from observations until the outputs exactly match reality. During this process, a conventional ANN can be prevented from making physically impossible predictions by imposing “soft constraints” on its outputs – worthwhile results that better conform to the laws of physics.

Beucler and colleagues used an ANN to simulate the climatic effects of atmospheric convection. Such models are only reliable if they adhere to the strict conservation of mass and energy. Instead of imposing these laws with soft constraints, the team embedded them as “hard constraints” in the architecture of the ANN. Comparable to the learning process of an infant, the method is synonymous with making predictions that defy physics literally unthinkable.

Adding these hard constraints makes it more difficult for the ANN to achieve its optimal output. The researchers say the price is well worth it, however, as conservation laws are critical to climate models.

–Marric Stephens

Marric Stephens is the corresponding editor for physics based in Bristol, UK.


  1. T. Beucler et al., “Enforcement of analytical constraints in neural networks that emulate physical systems”, Phys. Rev. Lett.126, 098302 (2021).

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