Multistable origami structures offer powerful opportunities for deployable and programmable mechanical systems, yet their design remains challenging due to strong geometric nonlinearity, coupled deformation modes, and the difficulty of shaping entire energy landscapes rather than isolated stable states. In particular, inverse design of multistable behavior typically requires extensive numerical datasets or full target energy curves, limiting generality, interpretability, and physical insight. In this work, we present a physics-informed neural network (PINN) framework for both forward prediction and inverse design of conical Kresling origami structures without relying on pre-collected training data. The proposed model embeds mechanical equilibrium and energy minimization principles directly into the learning process, ensuring physically admissible solutions by construction and suppressing non-physical artifacts. A truss-based energy formulation is used to describe the coupled axial and rotational mechanics, enabling the PINN to predict complete energy landscapes as a function of deformation. Beyond forward modeling, we introduce an inverse design strategy that programs multistable energy landscapes from minimal specifications. By prescribing only two stable heights and a target energy barrier, the framework autonomously infers the barrier location and reconstructs a full, smooth, and physically consistent energy curve. This approach significantly reduces the dimensionality of the design problem while retaining direct control over stability and deployment forces. The method is further extended to multi-layer origami assemblies, where layer-wise programming of energy barriers enables passive, sequential deployment without active control or feedback. The proposed framework is validated through finite element analysis and experiments on single-layer and multi-layer origami prototypes, demonstrating accurate reproduction of energy profiles, stability characteristics, and deployment sequences. These results establish a data-free, interpretable, and generalizable computational mechanics framework for energy-landscape programming of multistable origami and related metamaterial systems, with potential applications in deployable aerospace structures, soft robotics, and adaptive mechanical systems.