세계의 딥러닝 시장 규모는 2024년 309억 달러에 달했습니다. 향후 IMARC Group은 이 시장이 2033년까지 4,234억 달러에 달하고, 2025-2033년에 걸쳐 29.92%의 성장률(CAGR)을 나타낼 것으로 예측됩니다. 현재 북미가 시장을 독점하고 있으며 2024년에는 36.5% 이상의 큰 시장 점유율을 차지했습니다. 인공지능(AI) 도입 증가, 데이터 처리의 진보, 영상 인식과 음성 인식 수요 증가, 연구 개발(R&D)에 대한 투자, 빅 데이터와 클라우드 컴퓨팅 기술의 도입 등이 시장을 추진하는 주요인이 되고 있습니다.
시장은 주로 정보기술(IT) 산업의 대폭적인 확대에 의해 견인되고 있습니다. 또한 디지털화의 진전과 원시 데이터를 자동 추출하는 딥러닝의 보급이 시장 성장에 영향을 미치고 있습니다. 또한 사용 가능한 데이터를 자동으로 분석하여 데이터를 처리하므로 보다 효율적이고 정확한 의사 결정이 가능합니다. 또한 의료 서비스에서 사이버 보안, 사기 감지, 의료 영상 분석, 가상 환자 지원과 같은 광범위한 서비스 이용도 또 다른 큰 성장 촉진요인이되었습니다. 이 외에도 빅데이터 분석과 클라우드 컴퓨팅의 통합과 하드웨어 및 소프트웨어 처리를 개선하기 위한 지속적인 R&D(R&D)가 시장 성장을 더욱 가속화하고 있습니다. 게다가 이러한 기술이 제공하는 확장성과 컴퓨팅 능력을 통해 기업은 방대한 데이터 세트를 효율적으로 처리하고 분석할 수 있어 시장 전망은 밝습니다.
미국은 인공지능(AI) 기술의 급속한 발전과 AI 주도 연구개발에 대한 투자 증가로 주요 지역 시장으로 두드러지고 있습니다. 또한 복잡한 데이터로부터 실용적인 통찰력을 얻기 위한 고급 데이터 분석의 필요성은 특히 금융, 소매, 헬스케어 부문에서 성장의 주요 촉진요인이 되고 있습니다. 또한 AI 혁신을 장려하는 정부의 이니셔티브도 딥러닝이 자율 시스템과 스마트 디바이스에서 점점 더 많이 사용되고 있기 때문에 시장 성장을 더욱 뒷받침하고 있습니다. 2024년 11월 4일, Meta Platforms, Inc.는 미국 정부 기관과 국가 안보 계약자들이 회사의 인공지능 모델을 군사 용도로 이용할 것을 인정한다고 선언했습니다. 회사에 따르면, Lama라고 불리는 회사의 AI 모델을 연방 정부 기관이 사용할 수 있도록 한다고 합니다. 이 회사는 Lockheed Martin과 부즈 알렌과 같은 방위 관련 기업과 Parantier와 Andryl과 같은 방위를 전문으로하는 기술 기업과 협력하고 있습니다. 이 외에도 활발한 전자상거래 및 디지털 마케팅 부문은 맞춤형 고객 경험과 타겟 광고를 위해 딥러닝을 활용합니다. 또한 첨단 AI 솔루션 개발을 위한 첨단기술과 신흥기업의 제휴도 미국의 딥러닝 시장의 견조한 성장에 기여하고 있습니다.
이미지 인식 및 음성 인식에서 딥러닝 수요 증가
이미지내의 패턴이나 오브젝트, 특징을 분석·식별하는 수요 증가가 시장의 성장을 가속화하고 있습니다. 또한 딥러닝 기술을 기반으로 한 의료 영상 처리 솔루션은 비정상 감지 및 외과 수술 지원 기능과 같은 의료 부문의 애플리케이션과 함께 질병 진단 지원을 제공하므로 성장에 긍정적인 영향을 미칩니다. 이 외에도 이미지 인식 시스템은 교통 표지, 보행자 및 기타 장애물의 실시간 감지를 용이하게 하며, 자율주행 차량 감지 시 교통의 안전과 효율성 향상에 도움이 됩니다. 게다가 NLP 용도이나 음성 어시스턴트를 만드는데 꼭 필요한 음성 인식도 있습니다. 또한 딥러닝 모델은 음성을 텍스트로 쓰는 데 사용되며 Siri, Alexa, Google Assistant와 같은 음성 제어 가상 어시스턴트가 사용자의 명령을 정확하게 이해하고 응답할 수 있도록 합니다. 이를 통해 사람들의 기술과의 연관성이 바뀌어 핸즈프리로 직관적인 사용자 경험이 가능해졌습니다. 또한 고객서비스센터, 콜센터, 언어번역서비스에 음성 인식제품을 채택함으로써 커뮤니케이션을 간소화하고 응답시간을 향상시켜 시장 성장을 가속하고 있습니다.
R&D(R&D) 활동에 대한 투자 증가
딥러닝은 빠르게 진행되고 있으며, 다양한 산업 조직들이 이 기술의 기능과 응용을 개선하기 위해 많은 투자를 하고 있습니다. 게다가 R&D 투자는 성능, 정확성 및 효율성을 향상시키는 학습 측면과 새로운 알고리즘과 아키텍처 개발에 있어 시장 성장에 영향을 미칩니다. 또한 연구자들은 자연 언어 처리, 컴퓨터 비전 및 기타 AI 주도 작업에서 혁신을 달성하기 위해 주의 메커니즘, 변압기, 생성 적대 네트워크(GAN)와 같은 혁신적인 기술을 지속적으로 탐구하고 있습니다. 스탠포드 대학의 Artificial Index에 따르면, AI에 대한 민간 투자는 2023년에 전체적으로 감소했지만, 생성형 AI에 대한 자금 조달은 급격히 증가하여 2022년부터 거의 8배의 252억 달러에 달했습니다. Hugging Face, Inflection, Anthropic, OpenAI와 같은 유명한 생성형 AI 기업이 많은 돈을 모으는 라운드를 공개했습니다. 또한 하드웨어 최적화도 R&D 투자의 초점이 되고 있습니다. GPU(그래픽 프로세싱 유닛)와 TPU(텐서 프로세싱 유닛) 등 딥러닝 계산을 가속화하기 위해 설계된 전용 프로세서의 개발이 진행되고 있습니다. 이러한 하드웨어의 발전으로 학습 시간과 추론을 가속화할 수 있어 기업이 모델을 더욱 친숙하고 확장할 수 있습니다.
유리한 정부 이니셔티브 구현
정부의 지원과 이니셔티브는 시장 성장을 가속하는 데 필수적입니다. 게다가 정부는 인공지능(AI)의 변화 가능성을 인식하고 AI의 연구개발 프로젝트에 적극적으로 투자하고 연구개발을 촉진하여 시장 성장에 영향을 미치고 있습니다. 게다가 정부기관의 재정투자를 통해 대학, 연구기관, 비공개회사는 기술 혁신의 한계를 넓혀 기술적 진보를 촉진하는 야심적인 딥러닝 프로젝트를 실시할 수 있어 또 하나의 큰 성장유발요인이 되고 있습니다. 세계 정부의 이니셔티브는 딥러닝 비즈니스 확대에 박차를 가하고 있습니다. 예를 들어 유럽 연합의 Horizon Europe Program은 딥러닝과 인공지능 개발을 위해 934억 유로(980억 달러)를 할당하고 있습니다(2021-2027년). 미국의 국가 AI 이니셔티브 법은 AI의 R&D(R&D), 교육 및 표준 개발을 위한 자금을 늘리기 위해 5년간(2021-2026년) 약 65억 달러를 제공합니다. 반면 인도의 국가 AI 전략은 건강 관리, 교육 및 농업을 선호하고 2035년까지 GDP를 1조 달러로 끌어올릴 것으로 예측됩니다. 이러한 규정은 최첨단 딥러닝에 대한 국제 투자를 강조합니다.
이와는 별도로, 정부는 AI에 특화된 센터 오브 엑설런스나 혁신 허브를 설립하는 경향이 있으며, 이는 연구자, 학자, 산업 전문가를 위한 공동 공간이며, 지식 공유, 네트워킹, 학제간 연구를 촉진하고 딥러닝에서의 획기적인 발견을 조장하는 환경을 만들어 내고 있습니다. 또한 정부는 민간 파트너십에 적극적으로 참여하고 산업을 넘어 제품의 채택을 가속화하고 책임있는 AI의 개발과 전개를 장려하는 시책과 시책을 책정함으로써 시장 성장을 가속하고 있습니다.
The global deep learning market size reached USD 30.9 Billion in 2024. Looking forward, IMARC Group expects the market to reach USD 423.4 Billion by 2033, exhibiting a growth rate (CAGR) of 29.92% during 2025-2033. North America currently dominates the market, holding a significant market share of over 36.5% in 2024. The increasing artificial intelligence (AI) adoption, advancements in data processing, the growing demand for image and speech recognition, investments in research and development (R&D), and the introduction of big data and cloud computing technologies are some of the major factors propelling the market.
The market is primarily driven by the significant expansion of the information technology (IT) industry. In addition, the growing trend of digitalization, and the widespread adoption of deep learning for automatically extracting raw data, are influencing the market growth. It also processes data by automatically analyzing available data, resulting in more efficient and accurate decision-making. Moreover, the extensive service use of cybersecurity, fraud detection, medical image analysis, and virtual patient assistance in healthcare represents another major growth-inducing factor. Besides this, the integration of big data analytics and cloud computing and ongoing research and development (R&D) efforts to improve hardware and software processing are further accelerating the market growth. Furthermore, the scalability and computational power offered by these technologies allow organizations to process and analyze vast datasets efficiently, thus creating a positive market outlook.
The United States stands out as a key regional market, driven by rapid advancements in artificial intelligence (AI) technologies and increasing investments in AI-driven research and development. In addition, the need for sophisticated data analytics to yield actionable insights from complex data is another major driver of growth, especially in the finance, retail, and healthcare sectors. Government efforts to encourage AI innovation are also driving the market growth further, as deep learning is increasingly being used in autonomous systems and smart devices. On 4th November 2024, Meta Platforms, Inc. declared that it will allow U.S. government agencies and national security contractors to utilize its artificial intelligence models for military applications. The firm said it will make its AI models, which are called Llama, available to federal agencies. It is working with defense contractors such as Lockheed Martin and Booz Allen, as well as technology companies specializing in defense, such as Palantir and Anduril. Besides this, the flourishing e-commerce and digital marketing sectors are leveraging deep learning for personalized customer experiences and targeted advertising. Additionally, partnerships between tech giants and startups to develop cutting-edge AI solutions contribute to the robust growth of the deep learning market in the United States.
The rising demand for deep learning for image and speech recognition
The growing demand to analyse and identify patterns, objects, and features within images is escalating the market growth. Moreover, deep learning technology-based medical imaging solutions provide diagnostic support for diseases along with anomaly detection and supportive features in surgical procedures and other applications in the health department, thus impacting the growth positively. In addition to this, image recognition systems facilitate real-time detection of traffic signs, pedestrians, and other obstacles in the detection of autonomous vehicles that help increase road safety and efficiency of the same. In addition, there is speech recognition, which proves crucial in the making of NLP applications and a voice assistant. Also, deep learning models are employed to transcribe speech into text, enabling voice-controlled virtual assistants including Siri, Alexa, and Google Assistant to understand and respond to user commands accurately. This has transformed the way people interact with technology and enabled hands-free and intuitive user experiences. Furthermore, the product adoption of for speech recognition in customer service centers, call centers, and language translation services is streamlining communication and improving response times thus propelling the market growth.
The increasing investments in research and development (R&D) activities
Deep learning continues to advance at a rapid pace, and organizations in different industries are investing heavily in order to improve the capabilities and applications of this technology. Furthermore, investments in R&D are made on aspects of learning and the development of new algorithms and architectures that enhance performance, accuracy, and efficiency, thereby affecting market growth. Also, researchers are continuously exploring innovative techniques such as attention mechanisms, transformers, and generative adversarial networks (GANs) to achieve breakthroughs in natural language processing, computer vision, and other AI-driven tasks. According to the Artificial Index by Stanford University, private investment in AI fell overall in 2023, but financing for generative AI increased dramatically, almost octupling from 2022 to USD 25.2 Billion. Significant fundraising rounds were disclosed by prominent generative AI companies, such as Hugging Face, Inflection, Anthropic, and OpenAI. Moreover, hardware optimization is another focal point of R&D investments. Organizations are developing specialized processors, such as graphical processing units (GPUs) and tensor processing units (TPUs), designed to accelerate deep learning computations. These hardware advancements enable faster training times and inference, making the models more accessible and scalable for businesses.
The implementation of favorable government initiatives
Government support and initiatives are essential in fostering the market growth. Additionally, governments are recognizing the transformative potential of artificial intelligence (AI), and actively investing AI research and development projects, and promoting research, development, thus influencing market growth. Moreover, financial investments from government agencies allow universities, research institutions, and private companies to undertake ambitious deep-learning projects that push the boundaries of innovation and drive technological advancements representing another major growth-inducing factor. Global government initiatives are fuelling the expansion of the deep learning business. For instance, the Horizon Europe Program of the European Union allots €93.4 Billion (USD 98 Billion) (2021-2027) towards developments in deep learning and artificial intelligence. The U.S. National AI Initiative Act provides nearly USD 6.5 Billion over the five years (2021-2026) to increase funding for AI research and development (R&D), education, and standards development. In the meantime, India's National AI Strategy, which prioritises healthcare, education, and agriculture, is anticipated to boost GDP by USD 1 Trillion by 2035. These regulations highlight international investments in cutting-edge deep learning.
Apart from this, governments tend to create AI-focused centers of excellence and innovation hubs which are collaborative spaces for researchers, academics, and industry experts that facilitate knowledge sharing, networking, and interdisciplinary research, creating an environment that is conducive to breakthrough discoveries in deep learning. In addition, governments actively engage in public-private partnerships to accelerate the adoption of products across industries and create policies and regulations that encourage responsible AI development and deployment thus propelling the market growth.
Software leads the market with around 48.2% of market share in 2024. Software is crucial in the development and implementation of deep learning algorithms and models. It is a source that offers all the necessary tools and frameworks for researchers, data scientists, and developers to make complex neural networks and train them efficiently. Hence, software solutions have become the key to unlock the power of technology. Additionally, flexibility and scalability offered by the software make it highly attractive to businesses of all industries. Software-based solutions allow organizations to integrate deep learning capabilities into their existing systems and applications seamlessly, empowering businesses to use the power of AI-driven insights and automation to optimize processes, improve decision-making, and enhance customer experiences.
Besides this, the open-source nature of many software platforms fosters collaboration and knowledge sharing within the AI community. Popular open-source libraries such as TensorFlow and PyTorch are essential in democratizing access to technology, enabling widespread adoption and innovation. Furthermore, the continuous advancements in software, driven by ongoing research and development, are resulting in improved performance and efficiency.
Image recognition leads the market with around 40.5%of market share in 2024. Image recognition is currently dominating the market due to its wide-ranging applications and transformative impact across various industries. They are demonstrating exceptional capabilities in accurately identifying and analyzing objects, patterns, and features within images, making them highly sought after for diverse use cases. Moreover, deep learning-powered medical imaging systems aid in the early detection of diseases, assist in precise diagnoses, and support treatment planning in the healthcare industry.
Besides this, in the automotive sector, image recognition is essential for enabling advanced driver assistance systems (ADAS) and autonomous vehicles, enhancing safety and efficiency on the roads, thus accelerating the market growth. Moreover, the retail and e-commerce sectors use image recognition for visual search, product recommendation, and inventory management that enhances customer experiences, streamlines operations, and drives sales.
Security leads the market with around 12.8% of market share in 2024. Deep learning technology provides unparalleled capabilities in the detection, analysis, and response to sophisticated security breaches and attacks. Additionally, the growing demand for more powerful and sophisticated solutions to deal with the changing nature of cyber threats, is driving the market growth. In the cybersecurity domain, deep learning algorithms have an advantage over traditional security systems as they are efficient in detecting anomalies, patterns, and suspicious activities.
Moreover, the growing demand for cutting-edge security measures, such as deep learning-powered intrusion detection systems, malware detection, and behavioral analytics to offer organizations with enhanced defense mechanisms against emerging threats represents another major growth-inducing factor. Additionally, the vast amounts of data generated in the cybersecurity landscape require advanced data processing and analysis capabilities. It excels in handling big data and efficiently extracting meaningful insights, enabling security teams to make informed decisions and respond proactively to potential threats.
Recurrent neural networks (RNN) are designed to handle sequential data, such as time series or natural language. Their recurrent nature allows them to capture temporal dependencies within the data. RNNs have internal memory that enables them to process sequences of variable length, making them ideal for tasks such as language modeling, machine translation, and sentiment analysis.
In addition, CNNs are used for image and video processing tasks because they have the ability to extract features well using convolutional layers, which scan input data with small filters to identify patterns and spatial relationships. CNNs are widely used in image recognition, object detection, and image classification tasks because they can automatically learn relevant visual features. Apart from this, DBN stands for deep belief networks. These are generative models, consisting of multiple layers of stochastic, latent variables. They are used in unsupervised learning tasks, such as feature learning and dimensionality reduction. Hence, they find their use in applications such as speech recognition and recommendation systems.
Apart from this, deep stacking networks (DSN) are a type of autoencoder-based architecture used for unsupervised feature learning involving multiple stacked layers that progressively learn to encode and decode data representations which find applications in anomaly detection, data compression, and denoising tasks. Furthermore, gated recurrent units (GRU) are a variant of RNNs that aim to address the vanishing gradient problem and improve training efficiency which use gating mechanisms to regulate the flow of information through the network, allowing them to retain essential information for longer sequences and avoid long-term dependencies issues.
In 2024, North America accounted for the largest market share of over 36.5%. North America is home to some of the world's leading tech giants, research institutions, and AI startups, which heavily invest in research and development (R&D) for advanced technology. The presence of these industry leaders fosters a competitive ecosystem, driving advancements in algorithms, hardware, and software. Moreover, the highly skilled workforce comprising AI experts, data scientists, and engineers, is contributing to the development of sophisticated models and applications thus representing another major growth-inducing factor.
Besides this, North America's strong emphasis on entrepreneurship and venture capital funding allows the growth of AI-driven startups that often pioneer groundbreaking applications, further propelling market expansion. Additionally, supportive government policies, such as tax incentives and funding for AI research, encourage innovation, and attract businesses and investments to the region. Furthermore, the well-established infrastructure, including robust cloud computing services and high-performance computing resources, facilitates the scalability and deployment of complex deep learning models across the region.
United States Deep Learning Market Analysis
In 2024, US accounted for around 70.00% of the total North America deep learning market. Due to extensive use of machine learning applications, substantial investments in artificial intelligence (AI) research, and improvements in processing power, the US leads the world in the deep learning market. The US is a major leader in this technology. U.S.-based institutes produced 61 noteworthy AI models in 2023, significantly more than the European Union's 21 and China's 15. Innovation in this field is being led by companies such as Google, Microsoft, and NVIDIA, especially in areas like autonomous systems, computer vision, and natural language processing (NLP).
One of the main forces behind the advancements in drug development, personalised medicine, and diagnostics is the use of deep learning in healthcare. For instance, medical photographs may now be analysed with precision levels of 90% by incorporating deep learning algorithms. Deep learning is also being quickly incorporated into industries including finance, retail, and automotive for customer insights and predictive analytics. Big data's growth has also increased demand; according to current figures, IBM estimates that 2.5 quintillion bytes of data are created daily, which is so enormous that 90% of the world's data was created in the last two years. Accessibility is being further improved and market growth is being propelled by cloud-based platforms and the rise of AI-as-a-Service offerings by major providers.
Europe Deep Learning Market Analysis
The market for deep learning in Europe is growing because of its rich research infrastructure, strong government efforts, and growing industry use. To encourage the use of AI and deep learning, the European Union's Digital Europe Programme has set aside €7.5 Billion (Approximately USD 7.9 Billion) for 2021-2027, with a focus on applications in smart manufacturing, driverless cars, and healthcare. Additionally, The European Union plans to invest 1.4 Billion Euros (USD 1.5 Billion) to help the deep tech research industry in the region in the year 2025. The European Innovation Council (EIC), a division of the EU's research and innovation program, will provide the financing, which is an investment increase of around 200 million euros over 2024. Leading nations including the UK, France, and Germany are utilising deep learning for sophisticated robotics and industrial automation in accordance with Industry 4.0 objectives.
Major end use industries for this technology include the automotive and healthcare industries. In radiology and pathology, deep learning algorithms are frequently employed to increase diagnostic accuracy. Deep learning is being incorporated into self-driving technology in the automobile sector, with manufacturers such as Daimler and BMW making significant investments in AI-powered solutions. Furthermore, the use of deep learning to smart grids and renewable energy management has been accelerated by Europe's emphasis on sustainability. While Europe's strict data protection regulations, such as GDPR, have prompted the development of safe and moral AI frameworks, the expanding 5G infrastructure is also facilitating the adoption of edge AI solutions.
Asia Pacific Deep Learning Market Analysis
The deep learning market in Asia-Pacific is expanding at the quickest rate due to factors like growing investments in AI, rapid digitisation, and an increasingly tech-savvy populace. The top donors are India, South Korea, Japan, and China. The adoption of AI and generative AI technologies, such as software, services, and hardware made for AI-driven systems, is accelerating dramatically across the Asia/Pacific region. AI and Generative AI (GenAI) investments in the region are expected to reach USD 110 Billion by 2028, rising at a compound annual growth rate (CAGR) of 24.0% from 2023 to 2028, according to the most recent Worldwide AI and Generative AI Spending Guide published by International Development Corporation. The software and information services sector is one of the top adopters of AI, with a market share of 23.8% in 2024.
China's AI 2030 plan, which includes large investments in deep learning research, aims to establish the nation as a global leader in AI. With businesses like Toyota and Hyundai integrating AI in manufacturing and mobility solutions, South Korea and Japan are utilising deep learning in robots and autonomous vehicles. The proliferation of digital transactions and consumer data in India is propelling the use of deep learning in finance and e-commerce. Deep learning is also being used by the region's gaming and entertainment sectors to create immersive experiences and real-time personalisation.
Latin America Deep Learning Market Analysis
The growing adoption of AI and digital transformation across multiple industries is propelling the deep learning industry in Latin America. In the region, Brazil and Mexico are at the forefront in both application and investment. Deep learning is being applied in Brazil's vast agribusiness sector to improve productivity through crop monitoring and predictive analytics. Deep learning is being used in Mexico's retail and e-commerce sectors to forecast demand and gain insights into customers. Deep learning is also being used by the Latin American financial services industry for credit risk assessment and fraud detection, as fintech firms embrace AI-powered systems. Deep learning is also for identifying pavement failures in Latin American and the Caribbean. For instance, The Inter-American Development Bank (IDB) created the Pavimenta2 platform to evaluate road signage and to detect, monitor, and quantify pavement defects. Pavimenta2 uses computer vision technology, artificial intelligence (AI), and deep learning to automatically measure the locations and quantities of blurred lines, linear cracking, transversal cracking, crocodile cracking, rutting, and other failures by simply driving through the roadway network with a mounted cell phone or GoPro. The recorded video is then uploaded.
Middle East and Africa Deep Learning Market Analysis
The deep learning market in the Middle East and Africa (MEA) is in its initial stage but is witnessing rapid growth due to increasing investments in AI and smart city initiatives. With an emphasis on AI and deep learning technologies in Saudi Vision 2030 and Dubai's Smart City Strategy, nations like the United Arab Emirates and Saudi Arabia are leading the way in this adoption. Deep learning applications are also being used by the region's retail and healthcare industries to improve diagnostic precision and provide individualised services. For instance, AI-driven algorithms are being used by telemedicine companies in the United Arab Emirates to facilitate remote medical services. Additionally, the introduction of 5G networks and improvements in cloud infrastructure are enabling deep learning solutions to gain traction. The market is expected to pick up in the coming years. According to a survey conducted by Microsoft among AI leaders in 112 companies, across 7 sectors and 5 countries in the Middle East and Africa, it was found out that 89% of the respondents expect AI to generate business benefits by optimizing their companies' operations in the future.
At present, key players in the market are adopting various strategies to strengthen their position and gain a competitive edge. Companies are investing heavily in research and development (R&D) to stay at the forefront of deep learning technology focusing on improving algorithms, developing novel architectures, and exploring new applications to offer cutting-edge solutions to their customers. Moreover, several companies are engaging in strategic acquisitions and partnerships to expand their offerings and capabilities. Key players are expanding their operations to new geographic regions to tap into emerging markets and reach a broader customer base, including establishing regional offices, forming partnerships with local companies, and adapting their offerings to suit regional needs. They are providing excellent customer support and training services for customer satisfaction and loyalty and investing in customer support teams and educational resources to ensure their clients can maximize the value of their solutions.