Andraž Jelinčič

Andraž Jelinčič, Jiajie Tao, William F. Turner et al.

It is well understood that, when numerically simulating SDEs with general noise, achieving a strong convergence rate better than $O(\sqrt{h})$ (where h is the step size) requires the use of certain iterated integrals of Brownian motion, commonly referred to as its "Lévy areas". However, these stochastic integrals are difficult to simulate due to their non-Gaussian nature and for a $d$-dimensional Brownian motion with $d > 2$, no fast almost-exact sampling algorithm is known. In this paper, we propose LévyGAN, a deep-learning-based model for generating approximate samples of Lévy area conditional on a Brownian increment. Due to our "Bridge-flipping" operation, the output samples match all joint and conditional odd moments exactly. Our generator employs a tailored GNN-inspired architecture, which enforces the correct dependency structure between the output distribution and the conditioning variable. Furthermore, we incorporate a mathematically principled characteristic-function based discriminator. Lastly, we introduce a novel training mechanism termed "Chen-training", which circumvents the need for expensive-to-generate training data-sets. This new training procedure is underpinned by our two main theoretical results. For 4-dimensional Brownian motion, we show that LévyGAN exhibits state-of-the-art performance across several metrics which measure both the joint and marginal distributions. We conclude with a numerical experiment on the log-Heston model, a popular SDE in mathematical finance, demonstrating that high-quality synthetic Lévy area can lead to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC).

Andraz Jelincic, Ross C. Walker

The growing energy demand for computation is becoming increasingly unsustainable. Thermodynamic computing, which harnesses physical thermal fluctuations as a computational resource rather than suppressing them, offers orders-of-magnitude energy savings for probabilistic and combinatorial tasks. Pharmaceutical R&D, heavily reliant on computational optimization and sampling, is a natural application domain. Here we present what is, to our knowledge, the first concrete pharmaceutical application mapped to thermodynamic hardware with energy estimates grounded in prototype measurements. We reduce mRNA codon optimization, a combinatorial problem routinely solved in drug development, to sampling from an Ising model, making it directly executable on a thermodynamic sampling unit (TSU). Benchmarking three approaches (Potts sampling, Ising sampling, and a genetic algorithm baseline) on the SARS-CoV-2 spike protein, we find that all achieve comparable optimization quality (scores ~234-240), but energy estimates based on validated hardware models indicate that a TSU could solve this problem using approximately 10e6 times less energy than a conventional GPU. All code is released under an open-source license.

Andraž Jelinčič, James Foster, Patrick Kidger · oxford

Despite the success of adaptive time-stepping in ODE simulation, it has so far seen few applications for Stochastic Differential Equations (SDEs). To simulate SDEs adaptively, methods such as the Virtual Brownian Tree (VBT) have been developed, which can generate Brownian motion (BM) non-chronologically. However, in most applications, knowing only the values of Brownian motion is not enough to achieve a high order of convergence; for that, we must compute time-integrals of BM such as $\int_s^t W_r \, dr$. With the aim of using high order SDE solvers adaptively, we extend the VBT to generate these integrals of BM in addition to the Brownian increments. A JAX-based implementation of our construction is included in the popular Diffrax library (https://github.com/patrick-kidger/diffrax). Since the entire Brownian path produced by VBT is uniquely determined by a single PRNG seed, previously generated samples need not be stored, which results in a constant memory footprint and enables experiment repeatability and strong error estimation. Based on binary search, the VBT's time complexity is logarithmic in the tolerance parameter $\varepsilon$. Unlike the original VBT algorithm, which was only precise at some dyadic times, we prove that our construction exactly matches the joint distribution of the Brownian motion and its time integrals at any query times, provided they are at least $\varepsilon$ apart. We present two applications of adaptive high order solvers enabled by our new VBT. Using adaptive solvers to simulate a high-volatility CIR model, we achieve more than twice the convergence order of constant stepping. We apply an adaptive third order underdamped or kinetic Langevin solver to an MCMC problem, where our approach outperforms the No U-Turn Sampler, while using only a tenth of its function evaluations.