Marc Härkönen

Marc Härkönen, Markus Lange-Hegermann, Bogdan Raiţă

Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

Jianlei Huang, Marc Härkönen, Markus Lange-Hegermann et al.

Working with systems of partial differential equations (PDEs) is a fundamental task in computational science. Well-posed systems are addressed by numerical solvers or neural operators, whereas systems described by data are often addressed by PINNs or Gaussian processes. In this work, we propose Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs), a novel probabilistic framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions and can be conditioned on a finite data set. We explicitly construct GP priors for representative PDE systems with practical boundary conditions. Formal proofs of correctness are provided and empirical results demonstrating significant accuracy and computational resource improvements over state-of-the-art approaches.

Alexander Amini, Anna Banaszak, Harold Benoit et al.

We present LFM2, a family of Liquid Foundation Models designed for efficient on-device deployment and strong task capabilities. Using hardware-in-the-loop architecture search under edge latency and memory constraints, we obtain a compact hybrid backbone that combines gated short convolutions with a small number of grouped query attention blocks, delivering up to 2x faster prefill and decode on CPUs compared to similarly sized models. The LFM2 family covers 350M-8.3B parameters, including dense models (350M, 700M, 1.2B, 2.6B) and a mixture-of-experts variant (8.3B total, 1.5B active), all with 32K context length. LFM2's training pipeline includes a tempered, decoupled Top-K knowledge distillation objective that avoids support mismatch; curriculum learning with difficulty-ordered data; and a three-stage post-training recipe of supervised fine-tuning, length-normalized preference optimization, and model merging. Pre-trained on 10-12T tokens, LFM2 models achieve strong results across diverse benchmarks; for example, LFM2-2.6B reaches 79.56% on IFEval and 82.41% on GSM8K. We further build multimodal and retrieval variants: LFM2-VL for vision-language tasks, LFM2-Audio for speech, and LFM2-ColBERT for retrieval. LFM2-VL supports tunable accuracy-latency tradeoffs via token-efficient visual processing, while LFM2-Audio separates audio input and output pathways to enable real-time speech-to-speech interaction competitive with models 3x larger. LFM2-ColBERT provides a low-latency encoder for queries and documents, enabling high-performance retrieval across multiple languages. All models are released with open weights and deployment packages for ExecuTorch, llama.cpp, and vLLM, making LFM2 a practical base for edge applications that need fast, memory-efficient inference and strong task capabilities.