Toward a Physics-Informed TokaMind: Learning Magnetohydrodynamic Equilibrium Laws

Although foundation models for scientific domains promise scalable prediction across diverse modalities, they often lack explicit constraints to governing physical laws. In tokamak dynamics, this is critical since surrogates must predict magnetic equilibrium features essential for safe operation, but predictions from standard data-driven learning approaches are not guaranteed to respect the governing laws of plasma confinement (which is worsened by the scarcity of experimental fusion data compared to other domains). We present a methodology to augment the multi-modal foundation model TokaMind with physics-informed losses derived from the Grad-Shafranov equation, the cornerstone constraint on tokamak equilibrium. We introduce two loss formulations based on strong-form and weak-form residuals, and develop new evaluation metrics that additionally target physical consistency and operator-space accuracy. Fine-tuning experiments based on the TokaMark benchmark reveal that physics-informed regularization substantially reduces mutual consistency errors between predicted flux and current density, improves flux accuracy in operator space, and softens small-scale structural disruptions. These improvements come at a modest cost in signal-space normalized RMSE, indicating a fundamental trade-off between data fidelity and physical consistency. Our work connects foundation models and physical reasoning, providing both a reusable template for physics-informed training in scientific machine learning, and evidence that governing laws improve the reliability of equilibrium predictions for high-stakes plasma physics applications. Our open-sourced implementations will be made available upon acceptance.