Neural Policy Optimization

• Neural policy optimization is a framework that integrates neural networks with optimal control to generate scalable decision policies for high-dimensional systems.
• It combines model-based HJB residual minimization and actor-critic methods to balance sample efficiency, accuracy, and adaptability in various settings.
• Recent advances leverage system derivatives and physics-informed losses to enable real-time control with significantly reduced computational cost.

Neural policy optimization comprises algorithmic frameworks and theoretical results enabling the construction and training of neural network policies for decision and control problems, with a focus on settings where the system dynamics, objectives, or constraints induce high-dimensional or nonlinear relationships between state, control, and parameter spaces. Recent advances unify approaches from optimal control, dynamic programming, and reinforcement learning, leveraging neural networks both for value function approximation and direct policy representation in settings ranging from continuous deterministic systems to parameterized PDE-constrained and high-dimensional partially observed domains. Representative methodologies include (i) model-based approaches extracting feedback policies from value-function surrogates—especially via Hamilton-Jacobi-Bellman PDEs—and (ii) data-driven actor-critic or policy-gradient RL schemes learning from simulated or real system trajectories. This article synthesizes the algorithmic developments, performance tradeoffs, and empirical insights drawn from contemporary neural policy optimization research.

1. Foundations: Parameterized Optimal Control and Policy Representation

Neural policy optimization is motivated by the need to solve families of optimal control problems in which the system dynamics $f(x,u,\theta)$, running cost $\ell(x,u,\theta)$, and terminal cost $g(x_T,\theta)$ depend on parameters $\theta \in \Theta$. The state $x(t)\in\mathbb{R}^d$ evolves under

$\dot x(t) = f\bigl(x(t),u(t),\theta\bigr),$

with the objective

$$J(u;\theta)=\int_0^T \ell\bigl(x(t),u(t),\theta\bigr)\,dt + g\bigl(x(T),\theta\bigr).$$

A key goal is to amortize the solution—constructing a neural policy $\pi(x,t;\theta)$ that generalizes over $\theta$—so that control synthesis reduces to fast evaluation of a neural network, even for high-dimensional $x$ and $\theta$ (Verma et al., 2024).

The Hamilton–Jacobi–Bellman (HJB) equation provides a principle for feedback-law construction. The value function

$$V(x,t;\theta) = \inf_{u(\cdot)} J(u;\theta) \ \text{s.t.} \ \dot{x}=f(x,u,\theta),~x(0)=x$$

solves the PDE

$$\partial_t V + \min_u\bigl\{\ell(x,u,\theta)+\nabla_x V^\top f(x,u,\theta)\bigr\}=0,$$

terminalized by $V(x,T;\theta)=g(x,\theta)$. The optimal policy has feedback form: $$u^*(x,t;\theta)=\arg\min_u\bigl\{\ell(x,u,\theta)+\nabla_x V(x,t;\theta)^\top f(x,u,\theta)\bigr\}.$$ Neural networks enable scalable approximation of $V(\cdot)$ or direct parameter-to-action maps $\pi(\cdot)$ across high-dimensional configuration and parameter spaces.

2. Model-Based Neural Policy Optimization via HJB Residual Minimization

In the model-based paradigm, the value function $V(x,t;\theta)$ is represented by a neural network $\widehat{V}(x,t;\theta;w)$ (Verma et al., 2024). The training objective penalizes violations of the HJB equation along sampled trajectories: $$L_{\text{model}}(w) = \mathbb{E}_{x_0,\theta}\Bigl[ \int_0^T |\partial_t \widehat{V} + \min_u \{\ell + \nabla_x\widehat{V}^\top f\}|^2\,dt + \beta_2 |\widehat{V}(T) - g|^2 + \beta_3\|\nabla_x\widehat{V}(T) - \nabla_x g\|^2 \Bigr].$$ Feedback actions are extracted at each sample by

$$\hat u(x,t;\theta;w)=\arg\min_u\{\ell(x,u,\theta)+\nabla_x\widehat{V}(x,t;\theta;w)^\top f(x,u,\theta)\}.$$

Backpropagation computes gradients $$\partial_w\widehat{V},\,\partial_w\nabla_x\widehat{V},\,\partial_t\widehat{V}$$ through the network and the time integrator.

Empirically, the model-based approach scales to problems with $\dim(x)+\dim(\theta)\approx 1000$ using residual-style networks (width 64, depth 4) and demonstrates stable convergence due to explicit exploitation of system derivatives and physics-informed loss design. This significantly reduces the number of required environment (e.g., PDE) solves relative to data-driven RL, as demonstrated in high-dimensional advection–diffusion control (Verma et al., 2024).

3. Actor-Critic and Policy-Gradient Methods with Deep Neural Policies

In the data-driven RL paradigm, the policy (actor) $\pi(x,t;\theta;w_p)$ and value function (critic) $Q(x,u,t;\theta;w_c)$ are parametrized as neural networks—often convolutional for grid-structured or high-dimensional states. Training comprises:

• Critic update: Temporal-difference minimization,

$$L_c(w_c) = \mathbb{E}\Bigl[(r + \gamma Q(x',u';\theta;w_c^-) - Q(x,u;\theta;w_c))^2\Bigr],$$

with slow-moving target networks for stabilization.

• Actor update: Policy gradient or clipped surrogate loss (PPO/TD3),

$$L_a(w_p) = -\mathbb{E}\bigl[Q(x,\pi(x;\theta;w_p);\theta;w_c)\bigr] \ \text{or} \ \mathbb{E}\Bigl[\min\bigl(r_t(\theta)A_t,\mathrm{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\bigr)\Bigr],$$

where $r_t$ is the action likelihood ratio under new and old policies.

Modern implementations employ experience replay, entropy bonuses, target networks, and alternative value functions (e.g., GAE) to stabilize training for large networks and high-dimensional states.

The actor-critic trajectory efficiently explores the parameter/state/action space but is substantially more sample-intensive than the HJB-based method in settings where model derivatives are available. As demonstrated in (Verma et al., 2024), for a convection–diffusion control task, actor-critic methods required 15,000 environment solves and 5–7 hours of wall-clock time to reach moderate suboptimality ($>$0.2), while the model-based formulation achieved $<0.03$ suboptimality with $<1,000$ solves in $<40$ minutes.

4. Comparative Evaluation: Sample Complexity, Accuracy, and Scalability

The paradigm comparison on parameterized 2D PDE control reveals key trade-offs (Verma et al., 2024):

Method	#PDE Solves	Avg J	Subopt	Train Time
Model-based	800	0.15	0.02	30 min
PPO	15,000	0.28	0.22	6 h
TD3	15,000	0.26	0.20	5 h

For more complex parameterizations ("sinusoidal" case, $k=3$): | Method | #PDE Solves | Avg J | Subopt | Train Time | |-------------|------------:|------:|-------:|-----------:| | Model-based | 780 | 0.18 | 0.03 | 35 min | | PPO | 15,000 | 0.35 | 0.30 | 7 h | | TD3 | 15,000 | 0.32 | 0.27 | 6.5 h |

Key insights:

• Model-based neural HJB methods achieve higher accuracy with two orders of magnitude fewer environment solves and $10\times$ less wall-clock time.
• Actor-critic RL retains applicability when the model $f,\ell,g$ is unknown or derivative information is inaccessible, but requires extensive sampling and hyperparameter tuning.
• Model-based approaches require explicit model access and at least approximate analytical construction of the feedback law.

Scalability: The model-based approach remains tractable as the combined $n=\dim(x)+\dim(\theta)$ approaches $1,000$. RL approaches become prohibitive in sample complexity as $n\to 2,000$, primarily due to the rising cost of each environment solve.

5. Design Guidelines and Limitations

Empirical and theoretical analyses from (Verma et al., 2024) yield practical recommendations for neural policy optimization:

• Model-based methods: Use when system dynamics, costs, and derivatives are available and a feedback law can be constructed. Physics-informed losses, structured residual networks, and differentiation through the integrator fundamentally improve efficiency and solution quality.
• RL (actor-critic) methods: Apply in truly black-box or real-world settings where model structure is unavailable. Prepare for increased sample complexity and reliance on extensive hyperparameter search, experience replay, and potential instability.
• Network architecture: Employ residual-style architectures with smooth activations for value surrogates, convolutional stacks for high-dimensional spatial/pde states; leverage explicit encoding of time/parameters.
• Deployment: Amortized model-based training is ideal for high-fidelity simulators; learned networks yield real-time parameter-robust control.

Limitations:

• Model-based HJB approaches are inapplicable in the absence of system model or feedback law derivability.
• RL approaches incur high cost in settings where each environment interaction is expensive (e.g., when simulating complex PDEs).
• Both approaches can be bottlenecked by curse of dimensionality as the state/parameter grows, though model-based methods partially mitigate this via derivative structure.

6. Synthesis and Outlook

Neural policy optimization, uniting control-theoretic structure (value function residuals, feedback laws) and expressive neural parametrization, has enabled substantial advances in scalable, real-time solution of parameterized optimal control tasks. Model-based neural HJB frameworks attain much lower sample complexity and higher policy accuracy when system derivatives are accessible. Actor-critic RL remains a general-purpose tool for scenarios lacking model access, albeit at a substantial sampling and tuning cost.

This dual-path architecture—adopting model-based HJB when structure is available, and switching to actor-critic RL when required—constitutes an integrated toolkit for robust, parameter-aware neural policies in high-dimensional autonomous systems, control of PDEs, and rapid decision-making under uncertainty (Verma et al., 2024). The hybridization of control-theoretic residuals, differentiable program layers, and data-driven neural approximators suggests a promising research direction for real-time, scalable control in scientific, engineering, and large-scale cyber-physical domains.