Namkyeong Cho

Namkyeong Cho, Yeoneung Kim

We study policy iteration (PI) for deterministic infinite-horizon discounted optimal control problems, whose value function is characterized by a stationary Hamilton--Jacobi--Bellman (HJB) equation. At the PDE level, PI is fundamentally ill-posed: the improvement step requires pointwise evaluation of $\nabla V$, which is not well defined for viscosity solutions, and thus the associated nonlinear operator cannot be interpreted in a stable functional sense. We develop a monotone semi-discrete formulation for the stationary discounted setting by introducing a space-discrete scheme with artificial viscosity of order $O(h)$. This regularization restores comparison, ensures monotonicity of the discrete operator, and yields a well-defined pointwise policy improvement via discrete gradients. Our analysis reveals a convergence mechanism fundamentally different from the finite-horizon case. For each fixed mesh size $h>0$, we prove that the semi-discrete PI sequence converges monotonically and geometrically to the unique discrete solution, where the contraction is induced by the resolvent structure of the discounted operator. We further establish the sharp vanishing-viscosity estimate $\|V^h - V\|_{L^\infty} \leq C\sqrt{h}$, and derive a quantitative error decomposition that separates policy iteration error from discretization error, exhibiting a nontrivial coupling between iteration count and mesh size. Numerical experiments in nonlinear one and two-dimensional control problems confirm the theoretical predictions, including geometric convergence and the characteristic decay-then-plateau behavior of the total error.

Minseok Kim, Yeongjong Kim, Namkyeong Cho et al.

Physics-informed neural solvers offer a promising route to model-based reinforcement learning in continuous time, where optimal feedback synthesis is governed by Hamilton--Jacobi--Bellman (HJB) equations. Practical implementations often occupy a regime that is neither a classical grid method nor a continuous-PDE PINN: the value function is represented by a neural network, finite-difference HJB policy-evaluation operators are evaluated by network queries at shifted points, and residuals are minimized by random continuous collocation. This regime preserves the stabilized finite-difference policy-evaluation structure while avoiding grid-based value unknowns. We develop an error theory for this hybrid regime. Interpreting finite differences as shift operators acting on neural networks, we prove a population $L^2$ stability estimate for one policy-evaluation step with learned dynamics. The bound separates residual error, initial and exterior-collar mismatch, policy mismatch, and model-identification error, with an explicit gradient amplification factor for learned dynamics, while the underlying linear evaluation stability remains free of hidden inverse-viscosity blow-up. We further give a finite-sample collocation certificate and a conditional multi-step propagation result through greedy policy improvement. Experiments on compact-control LQR upto 64 dimensions, Allen--Cahn control, pendulum, Hopper, and 3D quadrotor benchmarks compare against representative model-based and model-free RL baselines, demonstrating the predicted residual, policy-mismatch, and learned-model error trends.

Namkyeong Cho, Yeoneung Kim

We address the crucial yet underexplored stability properties of the Hamilton--Jacobi--Bellman (HJB) equation in model-free reinforcement learning contexts, specifically for Lipschitz continuous optimal control problems. We bridge the gap between Lipschitz continuous optimal control problems and classical optimal control problems in the viscosity solutions framework, offering new insights into the stability of the value function of Lipschitz continuous optimal control problems. By introducing structural assumptions on the dynamics and reward functions, we further study the rate of convergence of value functions. Moreover, we introduce a generalized framework for Lipschitz continuous control problems that incorporates the original problem and leverage it to propose a new HJB-based reinforcement learning algorithm. The stability properties and performance of the proposed method are tested with well-known benchmark examples in comparison with existing approaches.

Yeongjong Kim, Yeoneung Kim, Minseok Kim et al.

We propose a physics-informed neural network policy iteration (PINN-PI) framework for solving stochastic optimal control problems governed by second-order Hamilton--Jacobi--Bellman (HJB) equations. At each iteration, a neural network is trained to approximate the value function by minimizing the residual of a linear PDE induced by a fixed policy. This linear structure enables systematic $L^2$ error control at each policy evaluation step, and allows us to derive explicit Lipschitz-type bounds that quantify how value gradient errors propagate to the policy updates. This interpretability provides a theoretical basis for evaluating policy quality during training. Our method extends recent deterministic PINN-based approaches to stochastic settings, inheriting the global exponential convergence guarantees of classical policy iteration under mild conditions. We demonstrate the effectiveness of our method on several benchmark problems, including stochastic cartpole, pendulum problems and high-dimensional linear quadratic regulation (LQR) problems in up to 10D.

Jaesun Shin, Eunjoo Jeon, Taewon Cho et al.

While message passing graph neural networks result in informative node embeddings, they may suffer from describing the topological properties of graphs. To this end, node filtration has been widely used as an attempt to obtain the topological information of a graph using persistence diagrams. However, these attempts have faced the problem of losing node embedding information, which in turn prevents them from providing a more expressive graph representation. To tackle this issue, we shift our focus to edge filtration and introduce a novel edge filtration-based persistence diagram, named Topological Edge Diagram (TED), which is mathematically proven to preserve node embedding information as well as contain additional topological information. To implement TED, we propose a neural network based algorithm, named Line Graph Vietoris-Rips (LGVR) Persistence Diagram, that extracts edge information by transforming a graph into its line graph. Through LGVR, we propose two model frameworks that can be applied to any message passing GNNs, and prove that they are strictly more powerful than Weisfeiler-Lehman type colorings. Finally we empirically validate superior performance of our models on several graph classification and regression benchmarks.

Namkyeong Cho, Junseung Ryu, Hyung Ju Hwang

This study investigates the impact of Sobolev Training on operator learning frameworks for improving model performance. Our research reveals that integrating derivative information into the loss function enhances the training process, and we propose a novel framework to approximate derivatives on irregular meshes in operator learning. Our findings are supported by both experimental evidence and theoretical analysis. This demonstrates the effectiveness of Sobolev Training in approximating the solution operators between infinite-dimensional spaces.