A singular function is a partial function such that at one or more points, the left and/or right limit diverge (e.g., the function 1/x). Since programming languages typically support division, programs may denote singular functions. Although on its own, a singularity may be considered a bug, introducing a division-by-zero error, singular integrals— a version of the integral that is well-defined when the integrand is a singular function and the domain of integration contains a singularity—arise in science and engineering, including in physics, aerodynamics, mechanical engineering, and computer graphics. In this paper, we present the first semantics of a programming language for singular integration. Our differentiable programming language, SingularFlow, supports the evaluation and differentiation of singular integrals. We formally define the denotational semantics of SingularFlow, deriving all the necessary mathematical machinery so that this work is rigorous and self-contained. We then define an operational semantics for SingularFlow that estimates integrals and their derivatives using Monte Carlo samples, and show that the operational semantics is a well-behaved estimator for the denotational semantics. We implement SingularFlow in JAX and evaluate the implementation on a suite of benchmarks that perform the finite Hilbert transform, an integral transform related to the Fourier transform, which arises in domains such as physics and electrical engineering. We then use SingularFlow to approximate the solutions to four singular integral equations— equations where the unknown function is in the integrand of a singular integral—arising in aerodynamics and mechanical engineering.

A *neural surrogate* is a neural network that mimics the behavior of a program. Neural surrogates of programs have been used to automatically tune program inputs, adapt programs to new settings, and accelerate computations. Neural surrogates have traditionally been developed by training on input-output examples for a single program. Language models present another approach wherein a model is trained on a single, large dataset then directly consumes program text, to act as a neural surrogate of the program. Having the language model as both the neural surrogate generator and the neural surrogate, however, poses a tradeoff of limited accuracy or excessive resource consumption. We present *neural surrogate compilation*, a technique for producing neural surrogates directly from program text without coupling neural surrogate generation and execution. We implement neural surrogate compilers using hypernetworks trained on a dataset of C programs and find they produce neural surrogates that are $1.91$-$9.50\times$ as data-efficient and train in $4.31$-$7.28\times$ fewer epochs than neural surrogates trained from scratch.

Computations in physical simulation, computer graphics, and probabilistic inference often require the differentiation of discontinuous processes due to contact, occlusion, and changes at a point in time. Popular differentiable programming languages, such as PyTorch and JAX, ignore discontinuities during differentiation. This is incorrect for parametric discontinuities —conditionals containing at least one real-valued parameter and at least one variable of integration. We introduce Potto, the first differentiable first-order programming language to soundly differentiate parametric discontinuities. We present a denotational semantics for programs and program derivatives and show the two accord. We describe the implementation of Potto, which enables separate compilation of programs. Our prototype implementation overcomes previous compile-time bottlenecks achieving an 88.1x and 441.2x speed up in compile time and a 2.5x and 7.9x speed up in runtime, respectively, on two increasingly large image stylization benchmarks. We showcase Potto by implementing a prototype differentiable renderer with separately compiled shaders.

Deep learning is moving towards increasingly sophisticated optimization objectives that employ higher-order functions, such as integration, continuous optimization, and root-finding. Since differentiable programming frameworks such as PyTorch and TensorFlow do not have first-class representations of these functions, developers must reason about the semantics of such objectives and manually translate them to differentiable code. We present a differentiable programming language, λ S , that is the first to deliver a semantics for higher-order functions, higher-order derivatives, and Lipschitz but nondifferentiable functions. Together, these features enableλ S to expose differentiable, higher-order functions for integration, optimization, and root-finding as first-class functions with automatically computed derivatives. λ S ’s semantics is computable, meaning that values can be computed to arbitrary precision, and we implement λ S as an embedded language in Haskell. We use λ S to construct novel differentiable libraries for representing probability distributions, implicit surfaces, and generalized parametric surfaces – all as instances of higher-order datatypes – and present case studies that rely on computing the derivatives of these higher-order functions and datatypes. In addition to modeling existing differentiable algorithms, such as a differentiable ray tracer for implicit surfaces, without requiring any user-level differentiation code, we demonstrate new differentiable algorithms, such as the Hausdorff distance of generalized parametric surfaces.

Emerging research in computer graphics, inverse problems, and machine learning requires us to differentiate and optimize parametric discontinuities. These discontinuities appear in object boundaries, occlusion, contact, and sudden change over time. In many domains, such as rendering and physics simulation, we differentiate the parameters of models that are expressed as integrals over discontinuous functions. Ignoring the discontinuities during differentiation often has a significant impact on the optimization process. Previous approaches either apply specialized hand-derived solutions, smooth out the discontinuities, or rely on incorrect automatic differentiation. We propose a systematic approach to differentiating integrals with discontinuous integrands, by developing a new differentiable programming language. We introduce integration as a language primitive and account for the Dirac delta contribution from differentiating parametric discontinuities in the integrand. We formally define the language semantics and prove the correctness and closure under the differentiation, allowing the generation of gradients and higher-order derivatives. We also build a system, Teg, implementing these semantics. Our approach is widely applicable to a variety of tasks, including image stylization, fitting shader parameters, trajectory optimization, and optimizing physical designs.

Algorithms for solid modeling, i.e., Computer-Aided Design (CAD) and computer graphics, are often specified on real numbers and then implemented with finite-precision arithmetic, such as floating-point. The result is that these implementations do not soundly compute the results that are expected from their specifications. We present a new library, StoneWorks, that provides sound and robust solid modeling primitives. We implement StoneWorks in MarshallB, a pure functional programming language for exact real arithmetic in which types denote topological spaces and functions denote continuous maps, ensuring that all programs are sound and robust. We developed MarshallB as an extension of the Marshall language. We also define a new shape representation, compact representation ( K-rep ), that enables constructions such as Minkowski sum and analyses such as Hausdorff distance that are not possible with traditional representations. K-rep is a nondeterminism monad for describing all the points in a shape. With our library, language, and representation together, we show that short StoneWorks programs can specify and execute sound and robust solid modeling algorithms and tasks.