To deal with the higher pileup after the high-luminosity upgrade of the LHC, the ATLAS liquid argon calorimeter readout electronics will be replaced. 556 Intel Agilex-7 FPGAs will be installed as part of the off-detector electronics, enabling a more sophisticated digital energy reconstruction. Instead of using a linear optimal filter like the current readout, we evaluated the use of recurrent and convolutional neural networks. These artificial neural networks are trained on signal-enriched simulated detector pulses for a set of representative cells. The network size is limited to approximately 500 multiply-accumulate operations per detector cell. They show improvements in energy reconstruction especially for the case of overlapping signal pulses, as they will appear more often under high-luminosity conditions. An FPGA prototype firmware has been developed for both RNNs and CNNs, that supports processing the required number of 384 detector cells on one FPGA. This uses a low-level VHDL approach to minimize the resource usage. The ANN architecture is fixed at compile time, but the weights are configurable per cell. The compilation results show that such a neural network based readout is feasible with the available hardware. This is possible through the use of time multiplexing, i.e. by running the FPGA at a multiple of the 40 MHz input data frequency.