Abstract
Scientific machine learning often requires combining known physics with unknown parameters or correction terms learned from data. Existing approaches either ignore known structure, encode it as a soft penalty, or require hand-written PyTorch code for each equation. We present The Neural Compiler, a system that translates programs written in a first-order Scheme-like expression language into frozen, differentiable PyTorch modules. These modules match the source program to floating-point precision and provide gradients through autograd. In hybrid models, the compiled module encodes known physics exactly while learned components model the unknown remainder. We evaluate the compiler across six experiment domains: Feynman physics equations, Lotka-Volterra dynamics, a damped pendulum, a one-dimensional heat equation, three-dimensional vector mechanics, and compositional generalization. Compiled modules match hand-coded PyTorch implementations numerically for single equations, showing no accuracy loss from compilation. With only 1 to 4 trainable parameters, compiled models recover physical constants to less than 1 percent error in most cases, while standard PINN baselines with more than 8500 parameters show 7 to 93 percent error. Compiled modules also compose with zero error, while neural approximations can accumulate large errors in deep composition chains. The main value of the compiler is not improved accuracy over hand-coded equations, but systematic composability: it generates correct, differentiable modules from symbolic specifications without rewriting each equation by hand. The system supports 51 primitive operations, including vector and matrix algebra, enabling PDE discretizations and hybrid scientific models. This string-in, module-out interface also provides a natural target for large language models that translate scientific descriptions into executable differentiable modules.
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The boundary between program execution and gradient-based optimization has long limited the use of code itself as a learnable scientific model. We present a compiler that translates a self-hosting subset of Scheme into differentiable computation graphs for autograd backends. Because the subset can compile its own evaluator, this yields differentiable meta-circular interpretation (DMCI): a compiled Scheme interpreter executes programs supplied as data, while reverse-mode autodiff propagates gradients to continuous constants embedded in those programs. The interpreter is compiled once, so new programs inherit differentiability without recompilation or custom gradient machinery, while retaining closures, recursion, and data structures. We prove that gradients through the compiled interpreter are correct almost everywhere and show that they match direct compilation to numerical precision across 171 recursive and higher-order program-seed pairs. We then use DMCI for program-and-parameter co-search, where a large language model proposes Scheme programs and exact gradients calibrate their continuous parameters through a single frozen interpreter. This enables OpenEvolve-style program search in which an outer loop proposes discrete program structures and DMCI supplies exact gradient-based calibration of each candidate's continuous parameters. On battery capacity-fade data, the search recovers a knee-like degradation structure and improves held-out extrapolation over hand-crafted baselines on the harder early-extrapolation split, matching them on the later split. On a high-dimensional El Nino inverse problem, DMCI optimizes an interpreted Kalman-filter likelihood where gradient-free search fails. These results extend symbolic regression and neurosymbolic search from closed-form expressions to executable, stateful programs, making model-generated code directly optimizable against data.
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