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I actually used the same noise algorithm for caves as i did for the surface terrain. Rather than 2d noise i used 3d noise. I just made a simple algorithm to increase the intensity of it based on how low the y position is. for the caves i just generated regular 3d noise across the map, and any place where the noise is greater than a certain value (I think i chose 0.8), it places an air block.

that is so cool i didnt know about 3d noise thats so cool my recreation is trash but you should check it out (i failed and did individual blocks instead of a actual terrain soo sorry)