I'm learning about AES, and I haven't been able to find any intuition or justification about why we expect AES to be difficult to break. How can we be confident that there aren't techniques for inferring the plaintext from the ciphertext, much more efficiently than brute-forcing the key? (And to be clear: I'm asking for an intuition, not a proof. I'm aware that we don't really have proof that any encryption scheme is unbreakable).
Many of the public-key protocols I've learned about are grounded in some simple, well-studied problem that we believe is fundamentally difficult. For example:
RSA: We believe RSA is secure because we believe it is fundamentally difficult to factor very large numbers.
Diffie-Hellman: We believe Diffie-Hellman is secure because we believe it is fundamentally difficult to compute the discrete logarithm.
Is there anything equivalent for AES?