Universal Approximation by Equivariant Message-Passing Graph Neural Networks with a Bilinear Readout

这个问题是要把给定的纯文本新闻正文排版成语义化的HTML片段，需要遵循几个规则。首先看正文内容，是一段关于BREMP网络的学术性内容，没有明显的小标题结构，所以可能不需要h2或h3划分。接下来看有没有列表、引用或代码，这段里没有这些元素，所以主要是处理段落。原文共五个自然段吗？不对，仔细看原文，其实原文是几个句子组成的几段？等一下，原文呈现的是：“We prove universal approximation...bilinear readout.”“Its layers combine...invariants from products of the equivariant features.”“We construct...same type.”“The construction uses...center.”“Iterating the construction...layer cutoff.”“The readout approximates...layer cutoff.”“We assume that...layers.”“Finally, DPA4...bilinear readout.”不对，再看用户给的正文，其实是： 第一段：We prove universal approximation for the bilinear-readout equivariant message-passing network (BREMP), a reference architecture for equivariant graph neural network interatomic potentials. Its layers combine equivariant convolutions with affine maps, and its readout forms invariants from products of the equivariant features. 第二段：We construct a one-layer feature that determines the local environment up to permutations of atoms of the same type. 第三段：The construction uses spherical harmonical filters...这里原文是“spherical harmonic filters”，对，第三段是这个。 第四段：Iterating the construction...layer cutoff. 第五段：The readout approximates...layer cutoff. 第六段：We assume that...layers. 第七段：Finally, DPA4...bilinear readout. 不对，再仔细看用户给的正文，其实是一段吗？不，看用户给的正文排版，其实是几个句子组成的段落？不对，用户给的正文是： “We prove universal approximation for the bilinear-readout equivariant message-passing network (BREMP), a reference architecture for equivariant graph neural network interatomic potentials. Its layers combine equivariant convolutions with affine maps, and its readout forms invariants from products of the equivariant features. We construct a one-layer feature that determines the local environment up to permutations of atoms of the same type. The construction uses spherical harmonic filters of degree no larger than the largest number of neighbors at a common distance from the center. Iterating the construction recovers the environment reachable in as many hops as there are layers, using the same bound on filter degrees at the layer cutoff. The readout approximates every continuous invariant function of the features uniformly on compact sets. Together these give universal approximation of atomic energies at a prescribed physical range using a shorter layer cutoff. We assume that, throughout the admissible class, message passing reaches all neighbors in each physical environment within a common number of layers. Finally, DPA4, an interatomic potential architecture in production use, inherits this result by approximating the BREMP layers constructed in the proof and realizing the bilinear readout.” 哦，原来用户给的正文是一段？不对，其实是几个自然段吗？看标点，其实这是一段连贯的内容，但用户可能希望每个自然段落用

包裹。等一下，再看规则1说每个自然段落用

包裹。那原文中的自然段落怎么划分？看原文的结构，第一句和第二句：“We prove...potentials. Its layers...features.” 这是一段吗？第三句：“We construct...type.” 是一段？第四句：“The construction uses...center.” 是一段？第五句：“Iterating the construction...cutoff.” 是一段？第六句：“The readout approximates...sets.” 是一段？第七句：“Together these give...cutoff.” 是一段？第八句：“We assume that...layers.” 是一段？第九句：“Finally...bilinear readout.” 是一段？不对，原文其实是： 看原文的句子： 1. We prove universal approximation for the bilinear-readout equivariant message-passing network (BREMP), a reference architecture for equivariant graph neural network interatomic potentials. 2. Its layers combine equivariant convolutions with affine maps, and its readout forms invariants from products of the equivariant features. 3. We construct a one-layer feature that determines the local environment up to permutations of atoms of the same type. 4. The construction uses spherical harmonic filters of degree no larger than the largest number of neighbors at a common distance from the center. 5. Iterating the construction recovers the environment reachable in as many hops as there are layers, using the same bound on filter degrees at the layer cutoff. 6. The readout approximates every continuous invariant function of the features uniformly on compact sets. 7. Together these give universal approximation of atomic energies at a prescribed physical range using a shorter layer cutoff. 8. We assume that, throughout the admissible class, message passing reaches all neighbors in each physical environment within a common number of layers. 9. Finally, DPA4, an interatomic potential architecture in production use, inherits this result by approximating the BREMP layers constructed in the proof and realizing the bilinear readout. 这九个句子，怎么划分段落？其实原文可能是一段，但用户给的正文里，有没有自然的段落划分？比如，第一句到第二句是介绍BREMP，第三句到第七句是构造和性质，第八句是假设，第九句是应用。但原文是连续的，没有分段标记。不对，看用户给的正文，其实是一段吗？不对，再看用户的问题中的正文，其实是： “We prove universal approximation for the bilinear-readout equivariant message-passing network (BREMP), a reference architecture for equivariant graph neural network interatomic potentials. Its layers combine equivariant convolutions with affine maps, and its readout forms invariants from products of the equivariant features. We construct a one-layer feature that determines the local environment up to permutations of atoms of the same type. The construction uses spherical harmonic filters of degree no larger than the largest number of neighbors at a common distance from the center. Iterating the construction recovers the environment reachable in as many hops as there are layers, using the same bound on filter degrees at the layer cutoff. The readout approximates every continuous invariant function of the features uniformly on compact sets. Together these give universal approximation of atomic energies at a prescribed physical range using a shorter layer cutoff. We assume that, throughout the admissible class, message passing reaches all neighbors in each physical environment within a common number of layers. Finally, DPA4, an interatomic potential architecture in production use, inherits this result by approximating the BREMP layers constructed in the proof and realizing the bilinear readout.” 哦，原来这是一段，但根据规则，每个自然段落用

包裹。那原文中的自然段落怎么分？其实原文中的每个句号后面的句子可能组成段落？但原文没有分段，所以可能用户希望把所有内容放在一个

里？不对，再看规则，规则1说每个自然段落用

包裹，所以需要看原文的自然段落。但原文是连续的，没有分段标记，所以可能用户给的正文是一段，所以用一个

包裹？但等一下，再仔细看，原文中的内容分为几个部分：1. 证明BREMP的通用近似；2. 构造一层特征；3. 构造的特征的性质；4. 读出的近似性；5. 假设；6. DPA4的应用。但原文是连续的，没有分段，所以可能用户希望把所有内容放在一个

里？不对，再看例子，比如如果原文有多个句子组成自然段，就用

包裹。那这段原文是一个自然段，所以用一个

包裹？然后有没有需要加的地方？规则3说重要句子可用。那哪些是重要句子？比如“We prove universal approximation for the bilinear-readout equivariant message-passing network (BREMP), a reference architecture for equivariant graph neural network interatomic potentials.” 这是核心结论，可以加；“Together these give universal approximation of atomic energies at a prescribed physical range using a shorter layer cutoff.” 这是结论，可以加；“Finally, DPA4, an interatomic potential architecture in production use, inherits this result by approximating the BREMP layers constructed in the proof and realizing the bilinear readout.” 这是应用，也可以加？或者看原文的逻辑，前两句是介绍BREMP，然后是构造，然后是性质，然后是结论，然后是假设，最后是应用。那现在需要把这些内容用

包裹，因为是一段。然后有没有需要加的地方？比如： 第一句：“We prove universal approximation for the bilinear-readout equivariant message-passing network (BREMP), a reference architecture for equivariant graph neural network interatomic potentials.” 是核心，可以加； 第二句：“Its layers combine equivariant convolutions with affine maps, and