arXiv subjects

Hong Chul Nam

Publications and source records attributed to Hong Chul Nam.

3 recordsLinked to original sources

arXiv2026

Operator Learning Using Weak Supervision from Walk-on-Spheres

Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) involving challenging optimization landscapes due to higher-order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk-on-Spheres method, we introduce a learning scheme called \emph{Walk-on-Spheres Neural Operator (WoS-NO)} which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data-free physics-informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre-computed datasets, avoids computing higher-order derivatives for loss functions that are memory-intensive and unstable, and demonstrates zero-shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75$\times$ improvement in $L_2$-error compared to standard physics-informed training schemes, up to 6.31$\times$ improvement in training speed, and reductions of up to 2.97$\times$ in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

cs.LG↗

arXiv2024

Solving Poisson Equations using Neural Walk-on-Spheres

We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. Leveraging stochastic representations and Walk-on-Spheres methods, we develop novel losses for neural networks based on the recursive solution of Poisson equations on spheres inside the domain. The resulting method is highly parallelizable and does not require spatial gradients for the loss. We provide a comprehensive comparison against competing methods based on PINNs, the Deep Ritz method, and (backward) stochastic differential equations. In several challenging, high-dimensional numerical examples, we demonstrate the superiority of NWoS in accuracy, speed, and computational costs. Compared to commonly used PINNs, our approach can reduce memory usage and errors by orders of magnitude. Furthermore, we apply NWoS to problems in PDE-constrained optimization and molecular dynamics to show its efficiency in practical applications.

cs.LG↗

arXiv2024

Victima: Drastically Increasing Address Translation Reach by Leveraging Underutilized Cache Resources

Address translation is a performance bottleneck in data-intensive workloads due to large datasets and irregular access patterns that lead to frequent high-latency page table walks (PTWs). PTWs can be reduced by using (i) large hardware TLBs or (ii) large software-managed TLBs. Unfortunately, both solutions have significant drawbacks: increased access latency, power and area (for hardware TLBs), and costly memory accesses, the need for large contiguous memory blocks, and complex OS modifications (for software-managed TLBs). We present Victima, a new software-transparent mechanism that drastically increases the translation reach of the processor by leveraging the underutilized resources of the cache hierarchy. The key idea of Victima is to repurpose L2 cache blocks to store clusters of TLB entries, thereby providing an additional low-latency and high-capacity component that backs up the last-level TLB and thus reduces PTWs. Victima has two main components. First, a PTW cost predictor (PTW-CP) identifies costly-to-translate addresses based on the frequency and cost of the PTWs they lead to. Second, a TLB-aware cache replacement policy prioritizes keeping TLB entries in the cache hierarchy by considering (i) the translation pressure (e.g., last-level TLB miss rate) and (ii) the reuse characteristics of the TLB entries. Our evaluation results show that in native (virtualized) execution environments Victima improves average end-to-end application performance by 7.4% (28.7%) over the baseline four-level radix-tree-based page table design and by 6.2% (20.1%) over a state-of-the-art software-managed TLB, across 11 diverse data-intensive workloads. Victima (i) is effective in both native and virtualized environments, (ii) is completely transparent to application and system software, and (iii) incurs very small area and power overheads on a modern high-end CPU.

cs.AR↗