The field of high-dimensional control currently faces a fundamental dichotomy: deep reinforcement learning (RL) offers massive scalability but remains opaque, functioning largely as a 'black box.' Conversely, classical control theory guarantees rigorous verification via the HJB equation yet collapses under the curse of dimensionality.

Our work addresses this dimensional gap by synthesizing the computational scalability of neural networks with the mathematical rigor of Pontryagin’s principles. Consequently, we aim to retain the model-free flexibility of deep learning while recovering the verifiable structure inherent in control theory.

Existing bridges, such as PINNs and Deep BSDEs, have attempted to recover structure by focusing primarily on the value function. However, in complex control landscapes, learning the global value function is often significantly harder than finding the optimal policy path itself.

We propose a policy-centric synthesis that bypasses this heavy computational overhead. This shift allows us to recover structural optimality by enforcing local optimality conditions, without the burden of mapping the entire value landscape. 

Conventional methodologies—namely standard RL, PINNs, and Deep BSDEs—often falter when capturing the intricate structures that frequently appear in high-dimensional control. Whether the difficulty stems from delay-driven path dependence, hard constraints that shatter gradients, or other structural irregularities, existing frameworks lack the native capacity to handle them effectively. 

Moreover, addressing certain structural features (e.g., time inconsistency or latent uncertainty) typically forces these methods into ill-fitting recursive approximations. PG-DPO overcomes these limitations by bypassing the need for a global recursive mold, instead aligning directly with local optimality conditions to robustly recover complex control structures. 

This schematic illustrates the core mechanism of our policy-centric pivot. Instead of following the traditional route—approximating the value function to apply gradient descent—we directly estimate the adjoint sensitivity.

Consequently, we compute the optimal control by strictly enforcing the Pontryagin Maximum Principle at every decision step. This paradigm shift moves us from merely learning a scalar value to learning the vector sensitivity, thereby enabling precise and verifiable control synthesis.

This comparative analysis delineates the structure-recovery capabilities across distinct challenging problem classes. While methodologies such as standard RL, PINNs, and Deep BSDEs often exhibit intrinsic limitations or necessitate elaborate adaptations to handle delays and hard constraints, PG-DPO accommodates these features natively.

By integrating constraints directly within the Hamiltonian maximization framework, our approach maintains robustness in regimes where alternative methods prove brittle, effectively transforming these traditionally 'non-native' impediments into tractable optimization components.

To empirically validate our framework, we present a representative case study: a high-dimensional constrained portfolio optimization involving 100 assets.

By leveraging barrier-regularized Hamiltonians, PG-DPO achieves linear scalability, maintaining high precision in a regime where traditional grid-based methods succumb to exponential failure.

Crucially, the results exhibit strictly zero constraint violations regarding short-selling. This demonstrates our core research achievement: we have successfully conquered high-dimensional complexity without sacrificing the structural rigor required to enforce hard constraints.

For those interested in further details, please refer to the document available at the following link. [Link] Thank you.