Dominant approaches for modelling Partial Differential Equations (PDEs) rely on deterministic predictions, yet many physical systems of interest are inherently chaotic and uncertain. While training probabilistic models from scratch is possible, it is computationally expensive and fails to leverage the significant resources already invested in high-performing deterministic backbones. In this work, we adopt a training-efficient strategy to transform pre-trained deterministic models into probabilistic ones via retrofitting with a proper scoring rule: the Continuous Ranked Probability Score (CRPS). Crucially, this approach is architecture-agnostic: it applies the same adaptation mechanism across distinct model backbones with minimal code modifications. The method proves highly effective across different scales of pre-training: for models trained on single dynamical systems, we achieve 20-54% reductions in rollout CRPS and up to 30% improvements in variance-normalised RMSE (VRMSE) relative to compute-matched deterministic fine-tuning. We further validate our approach on a PDE foundation model, trained on multiple systems and retrofitted on the dataset of interest, to show that our probabilistic adaptation yields an improvement of up to 40% in CRPS and up to 15% in VRMSE compared to deterministic fine-tuning. Validated across diverse architectures and dynamics, our results show that probabilistic PDE modelling need not require retraining from scratch, but can be unlocked from existing deterministic backbones with modest additional training cost.

翻译：当前对偏微分方程建模的主流方法依赖于确定性预测，然而许多感兴趣的物理系统本质上是混沌且不确定的。虽然从头训练概率模型是可行的，但计算成本高昂，且未能充分利用已投入高性能确定性主干网络的大量资源。本研究采用一种训练高效的策略，通过基于适当评分规则——连续排序概率评分——的改造，将预训练的确定性模型转化为概率模型。该方法的关键优势在于架构无关性：它通过最小限度的代码修改，即可在不同模型主干上应用相同的适应机制。该方法在不同预训练规模下均表现出显著效果：对于在单一动力系统上训练的模型，相比计算量匹配的确定性微调，我们在滚动CRPS上实现了20-54%的降低，在方差归一化均方根误差上获得了高达30%的提升。我们进一步在偏微分方程基础模型上验证了该方法——该模型在多个系统上预训练并在目标数据集上进行改造——结果表明，与确定性微调相比，我们的概率化适应在CRPS上带来高达40%的改进，在VRMSE上提升达15%。通过在多样化架构和动力学系统中的验证，我们的结果表明概率化偏微分方程建模无需从头开始重新训练，而是可以通过适度的额外训练成本，从现有确定性主干网络中解锁实现。