The equation of state (EOS) is essential for understanding material behavior under different pressure-temperature-volume (P-T-V) conditions across various disciplines. Traditional models, such as the Mie-Gr \(\ddot{\text {u}}\) neisen-Debye equation, rely on thermodynamic assumptions and expert knowledge, while classical Gaussian process based machine learning approaches can be sensitive to choice of kernels and are limited by scalability and extrapolability. To overcome these limitations, we propose EOSNN, a neural network based physics informed deep learning method that jointly learns multiple EOS surfaces from diverse data sources, including static and dynamic compression and ab initio calculations. Additionally, a probabilistic model is developed to account for both aleatoric and epistemic uncertainties. Our numerical experiments show that EOSNN outperforms traditional and Gaussian process methods in several aspects including accuracy, flexibility under different constraints, and extensibility for various tasks and constraints. Particularly on the challenging partially supervised task where energy information is limited on part of the Hugoniot curve, our method’s prediction for the energy off-Hugoniot can still reach a \(R^2\) score as high as 0.83 and RMSE as low as 0.52 eV/atom with proper selection of regularization for physical consistency. This result shows a significant improvement over the case without regularization, with an increase of 0.2 in \(R^2\) and a reduction of 0.26 eV/atom in energy prediction error, and is even slightly better than the fully supervised case for traditional regression method based on the Mie-Gr \(\ddot{\text {u}}\) neisen equation. These benefits can be further enhanced with physics-informed regularizations on quantities such as heat capacity ( \(C_V\) ), Gr \(\ddot{\text {u}}\) neisen parameter ( \(\gamma\) ) or bulk modulus ( \(K_T\) ).
