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Neural surrogate models are powerful and efficient tools in data mining. Meanwhile, large language models (LLMs) have demonstrated remarkable capabilities in code-related tasks. To this end, a novel application of LLMs emerges---using LLMs as surrogate models for code execution prediction. Given LLMs' unique ability to understand and process diverse programs, they present a promising direction for building general-purpose surrogate models. To systematically investigate this capability, we introduce SURGE, a comprehensive benchmark with 1160 problems covering 8 key aspects: multi-language programming tasks, competition-level programming problems, repository-level code analysis, high-cost scientific computing, time-complexity-intensive algorithms, buggy code analysis, programs dependent on specific compilers or execution environments, and formal mathematical proof verification. Through extensive analysis of 21 open-source and proprietary LLMs, we examine scaling laws, data efficiency, and predictive accuracy. Our findings reveal important insights about the feasibility of LLMs as efficient surrogates for computational processes.