Cell placement is an important phase of current VLSI circuit design styles such as standard cell, gate array, and Field Programmable Gate Array (FPGA). Although nondeterministic algorithms such as Simulated Annealing (SA) have been successful in solving this problem, they are known to be slow. In this paper, we propose a neural network algorithm that produces solutions as good as SA in substantially less time. Our algorithm is based on Mean Field Annealing (MFA) technique, which has been successfully applied to various combinatorial optimization problems. We derive a MFA formulation for the cell placement problem that can easily be applied to all VLSI design styles. To demonstrate that the proposed algorithm is applicable in practice, we derive a detailed formulation for the FPGA design style, and generate the layouts of several benchmark circuits.The performance of the proposed cell placement algorithm is evaluated in comparison with commercial automated circuit design software Xilinx Automatic Place and Route (APR) which uses SA technique. Performance evaluation is conducted using ACM/SIGDA Design Automation benchmark circuits. Experimental results indicate that the proposed MFA algorithm produces comparable results with APR. However, MFA is almost 20 times faster than APR on the average.