Are the rocks getting buried or are they sinking into the wet soil?

If it's burial, figure out where the soil is coming from. You may need to expand your river bed or perhaps add something to slow the water down.

If it's sinking, you could put down a geotextile fabric below the rocks to keep them atop the soil. Weed barrier fabric can be used in a pinch but I think a woven fabric to allow better water infiltration will be a better solution.

Do you have mulch there? Mulch would keep the soil from being "splashed" by raindrops - up and onto things, like rocks which are heavy enough to settle into mud.

Would a border between the bed and the lawn help direct the flow of water? It looks like it's washing from the grass at the bottom of the pic toward that corner of the bed. A brick/wood/rock border might help stop the water from flowing into the bed.

Have you though about putting a shallow trench right in front of the bed? Or a french drain under the rocks? I have done both and they both work with the shallow drain functioning as edging in looks. It has to be maintained once a year but no biggy. The french drain I covered with river rocks and it did clog after several years of record rains which washed part of my pasture into it.

The biggest question is where is the dirt coming from? The weed barrier under the rock will help with sinking as mentioned. If the soil is coming from the bed then you must stop the erosion at the source. Mulch, ground cover planting or weed barrier under mulch all come to mind. I have even had to put in gravel (rock) beds in very problematic areas, especially under the drip line of a house where there were no gutters. You could also use some sort of edging and take the slope away from the bed. Use geotextile fiber behind the edging.