High performance running shoes with ultralight elastomeric foams and carbon fiber plates have radically transformed the speed in long distance runnering. The elastomeric foam in the soles of the shoes stores energy during the stance phase and returns much of that energy during toe off. This can result in an up to 4% decrease in the metabolic cost of running at selected speeds [1]. In spite of the raising popularity of carbon fiber plate shoes among amateur and professional runners, the mechanical properties of the foams in these shoes have not been thoroughly investigated. Here we discover constitutive models for the two foams used in the Asics Metaspeed Sky Tokyo and Asics Metaspeed Edge Tokyo elite running shoes using constitutive neural networks. We performed uniaxial tension, uniaxial compression, confined compression, and torsion tests on foam samples over a wide stretch range of 0.4 - 1.3 and a shear strain range of 0 - 0.15. We found that the foam is highly compressible with an effective Poisson's ratio close to zero. By combining a variety of constitutive neural network architectures and sparse regression techniques [2], we found an interpretable model for each foam which fits all tests with an average R2 value of greater than 0.8. Our results shape the way towards finite element simulations of carbon fiber plate running shoes to improve performance and reduce metabolic cost. Our findings can be incorporated into biomechanical gait models to gain insights about personalized injury risk and performance improvement for an individual running in carbon fiber plated shoes. We expect that our new constitutive neural network architecture will generalize to automatically discover constitutive models for other highly compressible ultralight materials.