Tropical Geometry as a Restricted Architecture for Physics-Informed Neural Networks: Applications in Nonlinear Fluid-Structure Examples

What the math gives to ML

The paper's transferable contribution is a support-restricted representation: the trivial valuation maps a formal power series to the Boolean pattern of exponents whose coefficients are nonzero, while the t-adic order identifies its lowest active exponent. For polynomial differential equations, this suggests replacing an unconstrained PINN output with a basis containing only exponents admitted by tropical or formal-series analysis, making invalid singularity and asymptotic behaviors impossible rather than merely penalized. The most practical first test is a Taylor or local polynomial PINN for Van der Pol or Burgers, comparing a dense polynomial basis against a Boolean-masked basis generated from the residual's admissible exponent set.