Adrian M. Peter

Mawaba Pascal Dao, Adrian M. Peter

Active Inference, grounded in the Free Energy Principle, provides a powerful lens for understanding how agents balance exploration and goal-directed behavior in uncertain environments. Here, we propose a new planning framework, that integrates Monte Carlo Tree Search (MCTS) with active inference objectives to systematically reduce epistemic uncertainty while pursuing extrinsic rewards. Our key insight is that MCTS already renowned for its search efficiency can be naturally extended to incorporate free energy minimization by blending expected rewards with information gain. Concretely, the Cross-Entropy Method (CEM) is used to optimize action proposals at the root node, while tree expansions leverage reward modeling alongside intrinsic exploration bonuses. This synergy allows our planner to maintain coherent estimates of value and uncertainty throughout planning, without sacrificing computational tractability. Empirically, we benchmark our planner on a diverse set of continuous control tasks, where it demonstrates performance gains over both standalone CEM and MCTS with random rollouts.

Karthik S. Gurumoorthy, Adrian M. Peter, Birmingham Hang Guan et al.

We use a Schrödinger wave equation formalism to solve the eikonal equation. In our framework, a solution to the eikonal equation is obtained in the limit as Planck's constant $\hbar$ (treated as a free parameter) tends to zero of the solution to the corresponding linear Schrödinger equation. The Schrödinger equation corresponding to the eikonal turns out to be a \emph{generalized, screened Poisson equation}. Despite being linear, it does not have a closed-form solution for arbitrary forcing functions. We present two different techniques to solve the screened Poisson equation. In the first approach we use a standard perturbation analysis approach to derive a new algorithm which is guaranteed to converge provided the forcing function is bounded and positive. The perturbation technique requires a sequence of discrete convolutions which can be performed in $O(N\log N)$ using the Fast Fourier Transform (FFT) where $N$ is the number of grid points. In the second method we discretize the linear Laplacian operator by the finite difference method leading to a sparse linear system of equations which can be solved using the plethora of sparse solvers. The eikonal solution is recovered from the exponent of the resultant scalar field. Our approach eliminates the need to explicitly construct viscosity solutions as customary with direct solutions to the eikonal. Since the linear equation is computed for a small but non-zero $\hbar$, the obtained solution is an approximation. Though our solution framework is applicable to the general class of eikonal problems, we detail specifics for the popular vision applications of shape-from-shading, vessel segmentation, and path planning.